#set document(title: "1.2 Use the Language of Algebra", author: "OpenStax / XYZ Homework") #set page(width: 8.5in, height: auto, margin: 1in) #import "@preview/cetz:0.5.2" #set text(font: ("STIX Two Text", "Libertinus Serif", "New Computer Modern"), size: 10.5pt, lang: "en") #show math.equation: set text(font: ("STIX Two Math", "New Computer Modern Math")) #set par(justify: true, leading: 0.62em, spacing: 0.9em) #set enum(spacing: 1.1em) // room between list items so tall inline fractions don't collide #set list(spacing: 1.1em) #set table(stroke: 0.5pt + rgb("#c7ccd3")) #let BLUE = rgb("#183B6F") // brand navy — section bars + example/solution labels (white on navy 11.09:1) #let ORANGE = rgb("#A94509") // brand primary-700 — AA-safe deep orange for TEXT (5.93:1 on white; raw brand #F37021 is 2.94:1 and must never carry text) #let RED = rgb("#DC2626") // brand error-600 #let GREEN = rgb("#059669") // brand success-600 (decoration only; small green text uses green-text #007942) #show heading.where(level: 1): it => block(width: 100%, above: 0pt, below: 16pt, fill: gradient.linear(BLUE, rgb("#2C5AA0")), inset: (x: 14pt, y: 12pt), radius: 3pt, text(fill: white, weight: "bold", size: 19pt, it.body)) #show heading.where(level: 2): it => block(width: 100%, above: 18pt, below: 10pt, fill: BLUE, inset: (x: 10pt, y: 6pt), radius: 2pt, text(fill: white, weight: "bold", size: 12pt, it.body)) #show heading.where(level: 3): it => text(fill: ORANGE, weight: "bold", size: 12.5pt, it.body) #show heading.where(level: 4): it => text(fill: BLUE, weight: "bold", size: 10.5pt, it.body) #let examplebox(label, title, body) = block(width: 100%, breakable: true, fill: rgb("#EFF1F5"), stroke: 0.5pt + rgb("#CFDDF0"), radius: 4pt, inset: 10pt, above: 12pt, below: 12pt)[ #block(below: 6pt)[#box(fill: BLUE, inset: (x: 6pt, y: 2pt), radius: 2pt, text(fill: white, weight: "bold", size: 8.5pt, label)) #h(0.4em) #strong[#title]] #body] // rail = decorative left rule (raw brand token); labelcolor = AA-safe label text shade #let notebox(label, rail, labelcolor, tint, body) = block(width: 100%, breakable: true, fill: tint, stroke: (left: 3pt + rail), inset: (left: 10pt, rest: 8pt), radius: (right: 4pt), above: 11pt, below: 11pt)[ #text(fill: labelcolor, weight: "bold", size: 7.5pt, tracking: 0.5pt)[#upper(label)] #linebreak() #body] #let solutionbox(body) = block(above: 4pt, below: 8pt)[ #text(fill: BLUE, weight: "bold", size: 8.5pt)[Solution] #linebreak() #body] #let figph(msg) = block(width: 100%, height: 60pt, fill: rgb("#f6f7f9"), stroke: (paint: rgb("#c7ccd3"), dash: "dashed"), radius: 4pt, inset: 10pt)[ #align(center + horizon, text(fill: rgb("#889"), style: "italic", size: 9pt, msg))] // Standardize inlined figure sizes: measure the natural CeTZ canvas, then scale to a // consistent envelope (aspect-aware; see build_typst.py FIG_* constants). Unlike the // print preamble, dimensions are FLOORED: in an editor a user can trim a figure to a // degenerate 1-D shape (a bare line), and w/h or tw/w would then divide by zero. #let _STD_W = 3.5 #let _WIDE_W = 5.6 #let _MAX_H = 3.4 #let _ASPECT_WIDE = 2.2 #let _UPSCALE_MAX = 1.15 #let stdfig(body) = context { let m = measure(body) let w = calc.max(m.width / 1in, 0.01) let h = calc.max(m.height / 1in, 0.01) let tw = if w / h > _ASPECT_WIDE { _WIDE_W } else { _STD_W } let s = calc.min(tw / w, _MAX_H / h, _UPSCALE_MAX) align(center, box(scale(x: s * 100%, y: s * 100%, reflow: true, body))) } #show figure: set block(breakable: false) #set figure(gap: 8pt) #show figure.caption: set text(size: 8.5pt, fill: rgb("#555")) == 1.2#h(0.6em)Use the Language of Algebra #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ A more thorough introduction to the topics covered in this section can be found in the #emph[Prealgebra] chapter, #strong[The Language of Algebra]. ] === Use Variables and Algebraic Symbols Suppose this year Greg is 20 years old and Alex is 23. You know that Alex is 3 years older than Greg. When Greg was 12, Alex was 15. When Greg is 35, Alex will be 38. No matter what Greg’s age is, Alex’s age will always be 3 years more, right? In the language of algebra, we say that Greg’s age and Alex’s age are #strong[variable]#strong[s] and the 3 is a #strong[constant]. The ages change (“vary”) but the 3 years between them always stays the same (“constant”). Since Greg’s age and Alex’s age will always differ by 3 years, 3 is the #emph[constant]. In algebra, we use letters of the alphabet to represent variables. So if we call Greg’s age #emph[g], then we could use #math.equation(block: false, alt: "g plus 3")[$g + 3$] to represent Alex’s age. #figure(table( columns: 2, align: left, inset: 6pt, table.header([Greg’s age], [Alex’s age]), [#math.equation(block: false, alt: "12")[$12$]], [#math.equation(block: false, alt: "15")[$15$]], [#math.equation(block: false, alt: "20")[$20$]], [#math.equation(block: false, alt: "23")[$23$]], [#math.equation(block: false, alt: "35")[$35$]], [#math.equation(block: false, alt: "38")[$38$]], [#math.equation(block: false, alt: "g")[$"g"$]], [#math.equation(block: false, alt: "g plus 3")[$"g" + 3$]], )) The letters used to represent these changing ages are called #emph[variables]. The letters most commonly used for variables are #emph[x], #emph[y], #emph[a], #emph[b], and #emph[c]. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Variable] A #strong[variable] is a letter that represents a number whose value may change. ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Constant] A #strong[constant] is a number whose value always stays the same. ] To write algebraically, we need some operation symbols as well as numbers and variables. There are several types of symbols we will be using. There are four basic arithmetic #strong[operations]: #strong[addition], #strong[subtraction], #strong[multiplication], and #strong[division]. We’ll list the symbols used to indicate these operations in the table below. You’ll probably recognize some of them. #figure(table( columns: 4, align: left, inset: 6pt, table.header([Operation], [Notation], [Say:], [The result is…]), [Addition], [#math.equation(block: false, alt: "a plus b")[$a + b$]], [#emph[a] plus #emph[b]], [the sum of #emph[a] and #emph[b]], [Subtraction], [#math.equation(block: false, alt: "a minus b")[$a − b$]], [#emph[a] minus #emph[b]], [the difference of #emph[a] and #emph[b]], [Multiplication], [#math.equation(block: false, alt: "a times b , a b , open parenthesis a close parenthesis open parenthesis b close parenthesis ,")[$a · b , a b , ( a ) ( b ) ,$] #math.equation(block: false, alt: "open parenthesis a close parenthesis b , a open parenthesis b close parenthesis")[$( a ) b , a ( b )$]], [#emph[a] times #emph[b]], [the product of #emph[a] and #emph[b]], [Division], [#math.equation(block: false, alt: "a divided by b , a / b , the fraction a over b , b a")[$a ÷ b , a "/" b , frac(a, b) , b ") " overline(a)$]], [#emph[a] divided by #emph[b]], [the quotient of #emph[a] and #emph[b], #emph[a] is called the dividend, and #emph[b] is called the divisor], )) We perform these operations on two numbers. When translating from symbolic form to English, or from English to symbolic form, pay attention to the words “of” and “and.” - The #emph[difference of] 9 and 2 means subtract 9 and 2, in other words, 9 minus 2, which we write symbolically as #math.equation(block: false, alt: "9 minus 2 .")[$9 − 2 .$] - The #emph[product of] 4 and 8 means multiply 4 and 8, in other words 4 times 8, which we write symbolically as #math.equation(block: false, alt: "4 times 8 .")[$4 · 8 .$] In algebra, the cross symbol, #math.equation(block: false, alt: "times ,")[$× ,$] is not used to show multiplication because that symbol may cause confusion. Does 3#emph[xy] mean #math.equation(block: false, alt: "3 times y")[$3 #h(0.2em) × #h(0.2em) y$] (‘three times #emph[y]’) or #math.equation(block: false, alt: "3 times x times y")[$3 · x · y$] (three times #emph[x] times #emph[y])? To make it clear, use #math.equation(block: false, alt: "times")[$·$] or parentheses for multiplication. When two quantities have the same value, we say they are equal and connect them with an #strong[equal sign]. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Equality Symbol] #math.equation(block: false, alt: "a equals b")[$a = b$] is read “#emph[a] is equal to #emph[b]” The symbol #math.equation(block: false, alt: "“=”")[$“=”$] is called the #strong[equal sign]. ] On the number line, the numbers get larger as they go from left to right. The number line can be used to explain the symbols “\<” and “\>.” #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Inequality] #math.equation(block: true, alt: "a less than b is read “ a is less than b ”; a is to the left of b on the number line")[$a < b #h(0.2em) "is read “" a #h(0.2em) "is less than" #h(0.2em) b "”" \ a #h(0.2em) "is to the left of" #h(0.2em) b #h(0.2em) "on the number line"$]#figure(figph[A number line with points a and b labeled.], alt: "A number line with points a and b labeled.", caption: none) #math.equation(block: true, alt: "a greater than b is read “ a is greater than b ”; a is to the right of b on the number line")[$a > b #h(0.2em) "is read “" a #h(0.2em) "is greater than" #h(0.2em) b "”" \ a #h(0.2em) "is to the right of" #h(0.2em) b #h(0.2em) "on the number line"$]#figure(figph[A numbered line with points b and a labeled.], alt: "A numbered line with points b and a labeled.", caption: none) ] The expressions #emph[a] \< b or a \> #emph[b] can be read from left to right or right to left, though in English we usually read from left to right . In general, #emph[a] \< b is equivalent to b \> #emph[a]. For example 7 \< 11 is equivalent to 11 \> 7. And #emph[a] \> #emph[b] is equivalent to #emph[b] \< a. For example 17 \> 4 is equivalent to 4 \< 17. #figure(table( columns: 2, align: left, inset: 6pt, table.header([Inequality Symbols], [Words]), [#math.equation(block: false, alt: "a not equal to b")[$a ≠ b$]], [#emph[a] is #emph[not equal to] #emph[b]], [#emph[a] \< #emph[b]], [#emph[a] is #emph[less than] #emph[b]], [#math.equation(block: false, alt: "a less than or equal to b")[$a ≤ b$]], [#emph[a] is #emph[less than or equal to] #emph[b]], [#emph[a \> b]], [#emph[a] is #emph[greater than] #emph[b]], [#math.equation(block: false, alt: "a greater than or equal to b")[$a ≥ b$]], [#emph[a] is #emph[greater than or equal to] #emph[b]], )) #examplebox("Example 1")[][ Translate from algebra into English: ⓐ #math.equation(block: false, alt: "17 less than or equal to 26")[$17 ≤ 26$] ⓑ #math.equation(block: false, alt: "8 not equal to 17 minus 3")[$8 ≠ 17 − 3$] ⓒ #math.equation(block: false, alt: "12 greater than 27 divided by 3")[$12 > 27 ÷ 3$] ⓓ #math.equation(block: false, alt: "y plus 7 less than 19")[$y + 7 < 19$] #solutionbox[ ⓐ #math.equation(block: false, alt: "17 less than or equal to 26")[$17 ≤ 26$] #linebreak() 17 is less than or equal to 26 #linebreak() #linebreak() ⓑ #math.equation(block: false, alt: "8 not equal to 17 minus 3")[$8 ≠ 17 − 3$] #linebreak() 8 is not equal to 17 minus 3 #linebreak() #linebreak() ⓒ #math.equation(block: false, alt: "12 greater than 27 divided by 3")[$12 > 27 ÷ 3$] #linebreak() 12 is greater than 27 divided by 3 #linebreak() #linebreak() ⓓ #math.equation(block: false, alt: "y plus 7 less than 19")[$y + 7 < 19$] #linebreak() #emph[y] plus 7 is less than 19 ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Translate from algebra into English: ⓐ #math.equation(block: false, alt: "14 less than or equal to 27")[$14 ≤ 27$] ⓑ #math.equation(block: false, alt: "19 minus 2 not equal to 8")[$19 − 2 ≠ 8$] ⓒ #math.equation(block: false, alt: "12 greater than 4 divided by 2")[$12 > 4 ÷ 2$] ⓓ #math.equation(block: false, alt: "x minus 7 less than 1")[$x − 7 < 1$] #solutionbox[ ⓐ 14 is less than or equal to 27 ⓑ 19 minus 2 is not equal to 8 ⓒ 12 is greater than 4 divided by 2 ⓓ #emph[x] minus 7 is less than 1 ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Translate from algebra into English: ⓐ #math.equation(block: false, alt: "19 greater than or equal to 15")[$19 ≥ 15$] ⓑ #math.equation(block: false, alt: "7 equals 12 minus 5")[$7 = 12 − 5$] ⓒ #math.equation(block: false, alt: "15 divided by 3 less than 8")[$15 ÷ 3 < 8$] ⓓ #math.equation(block: false, alt: "y plus 3 greater than 6")[$y + 3 > 6$] #solutionbox[ ⓐ 19 is greater than or equal to 15 ⓑ 7 is equal to 12 minus 5 ⓒ 15 divided by 3 is less than 8 ⓓ #emph[y] plus 3 is greater than 6 ] ] Grouping symbols in algebra are much like the commas, colons, and other punctuation marks in English. They help to make clear which expressions are to be kept together and separate from other expressions. We will introduce three types now. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Grouping Symbols] #math.equation(block: true, alt: "Parentheses, open parenthesis close parenthesis; Brackets, [ ]; Braces, { }")[$"Parentheses" & & & #h(0.15em) \ "Brackets" & & & #h(0.15em) [ ] \ "Braces" & & & \{ \}$] ] Here are some examples of expressions that include #strong[grouping symbols]. We will simplify expressions like these later in this section. #math.equation(block: true, alt: "8 open parenthesis 14 minus 8 close parenthesis, 21 minus 3 [ 2 plus 4 open parenthesis 9 minus 8 close parenthesis ], 24 divided by { 13 minus 2 [ 1 open parenthesis 6 minus 5 close parenthesis plus 4 ] }")[$8 ( 14 − 8 ) & & & 21 − 3 [ 2 + 4 ( 9 − 8 ) ] & & & 24 ÷ \{ 13 − 2 [ 1 ( 6 − 5 ) + 4 ] \}$]What is the difference in English between a phrase and a sentence? A phrase expresses a single thought that is incomplete by itself, but a sentence makes a complete statement. “Running very fast” is a phrase, but “The football player was running very fast” is a sentence. A sentence has a subject and a verb. In algebra, we have #emph[expressions] and #emph[equations]. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Expression] An #strong[expression] is a number, a variable, or a combination of numbers and variables using operation symbols. ] An #strong[expression] is like an English phrase. Here are some examples of expressions: #figure(table( columns: 3, align: left, inset: 6pt, table.header([Expression], [Words], [English Phrase]), [#math.equation(block: false, alt: "3 plus 5")[$3 + 5$]], [3 plus 5], [the sum of three and five], [#math.equation(block: false, alt: "n minus 1")[$n − 1$]], [#emph[n] minus one], [the difference of #emph[n] and one], [#math.equation(block: false, alt: "6 times 7")[$6 · 7$]], [6 times 7], [the product of six and seven], [#math.equation(block: false, alt: "the fraction x over y")[$frac(x, y)$]], [#emph[x] divided by #emph[y]], [the quotient of #emph[x] and #emph[y]], )) Notice that the English phrases do not form a complete sentence because the phrase does not have a verb. An #strong[equation] is two expressions linked with an #strong[equal sign]. When you read the words the symbols represent in an equation, you have a complete sentence in English. The equal sign gives the verb. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Equation] An #strong[equation] is two expressions connected by an equal sign. ] Here are some examples of equations. #figure(table( columns: 2, align: left, inset: 6pt, table.header([Equation], [English Sentence]), [#math.equation(block: false, alt: "3 plus 5 equals 8")[$3 + 5 = 8$]], [The sum of three and five is equal to eight.], [#math.equation(block: false, alt: "n minus 1 equals 14")[$n − 1 = 14$]], [#emph[n] minus one equals fourteen.], [#math.equation(block: false, alt: "6 times 7 equals 42")[$6 · 7 = 42$]], [The product of six and seven is equal to forty-two.], [#math.equation(block: false, alt: "x equals 53")[$x = 53$]], [#emph[x] is equal to fifty-three.], [#math.equation(block: false, alt: "y plus 9 equals 2 y minus 3")[$y + 9 = 2 y − 3$]], [#emph[y] plus nine is equal to two #emph[y] minus three.], )) #examplebox("Example 2")[][ Determine if each is an expression or an equation: ⓐ #math.equation(block: false, alt: "2 open parenthesis x plus 3 close parenthesis equals 10")[$2 ( x + 3 ) = 10$] ⓑ #math.equation(block: false, alt: "4 open parenthesis y minus 1 close parenthesis plus 1")[$4 ( y − 1 ) + 1$] ⓒ #math.equation(block: false, alt: "x divided by 25")[$x ÷ 25$] ⓓ #math.equation(block: false, alt: "y plus 8 equals 40")[$y + 8 = 40$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([ⓐ #math.equation(block: false, alt: "2 open parenthesis x plus 3 close parenthesis equals 10")[$2 ( x + 3 ) = 10$]], [This is an #emph[equation]—two expressions are connected with an equal sign.]), [ⓑ #math.equation(block: false, alt: "4 open parenthesis y minus 1 close parenthesis plus 1")[$4 ( y − 1 ) + 1$]], [This is an #emph[expression]—no equal sign.], [ⓒ #math.equation(block: false, alt: "x divided by 25")[$x ÷ 25$]], [This is an #emph[expression]—no equal sign.], [ⓓ #math.equation(block: false, alt: "y plus 8 equals 40")[$y + 8 = 40$]], [This is an #emph[equation]—two expressions are connected with an equal sign.], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Determine if each is an expression or an equation: ⓐ #math.equation(block: false, alt: "3 open parenthesis x minus 7 close parenthesis equals 27")[$3 ( x − 7 ) = 27$] ⓑ #math.equation(block: false, alt: "5 open parenthesis 4 y minus 2 close parenthesis minus 7")[$5 ( 4 y − 2 ) − 7$]. #solutionbox[ ⓐ equation ⓑ expression ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Determine if each is an expression or an equation: ⓐ #math.equation(block: false, alt: "y cubed divided by 14")[$y^(3) ÷ 14$] ⓑ #math.equation(block: false, alt: "4 x minus 6 equals 22")[$4 x − 6 = 22$]. #solutionbox[ ⓐ expression ⓑ equation ] ] Suppose we need to multiply 2 nine times. We could write this as #math.equation(block: false, alt: "2 times 2 times 2 times 2 times 2 times 2 times 2 times 2 times 2 .")[$2 · 2 · 2 · 2 · 2 · 2 · 2 · 2 · 2 .$] This is tedious and it can be hard to keep track of all those 2s, so we use exponents. We write #math.equation(block: false, alt: "2 times 2 times 2")[$2 · 2 · 2$] as #math.equation(block: false, alt: "2 cubed")[$2^(3)$] and #math.equation(block: false, alt: "2 times 2 times 2 times 2 times 2 times 2 times 2 times 2 times 2")[$2 · 2 · 2 · 2 · 2 · 2 · 2 · 2 · 2$] as #math.equation(block: false, alt: "2 to the power 9 .")[$2^(9) .$] In expressions such as #math.equation(block: false, alt: "2 cubed ,")[$2^(3) ,$] the 2 is called the #emph[base] and the 3 is called the #emph[exponent]. The exponent tells us how many times we need to multiply the base. #figure(figph[The number two is shown with a superscipted number three to the right of it. an arrow is drawn to the number two and labeled “base” while another arrow is drawn to the superscripted three and labeled “exponent”. This means multiply 2 by itself, three times, as in 2 times 2 times 2.], alt: "The number two is shown with a superscipted number three to the right of it. an arrow is drawn to the number two and labeled “base” while another arrow is drawn to the superscripted three and labeled “exponent”. This means multiply 2 by itself, three times, as in 2 times 2 times 2.", caption: none) We read #math.equation(block: false, alt: "2 cubed")[$2^(3)$] as “two to the third power” or “two cubed.” We say #math.equation(block: false, alt: "2 cubed")[$2^(3)$] is in #emph[exponential notation] and #math.equation(block: false, alt: "2 times 2 times 2")[$2 · 2 · 2$] is in #emph[expanded notation]. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Exponential Notation] #math.equation(block: false, alt: "a to the power n")[$a^(n)$] means multiply #emph[a] by itself, #emph[n] times. #figure(figph[a is shown with a superscripted n to the right of it. an arrow is drawn to a and labeled “base” while another arrow is drawn to the superscripted n and labeled “exponent”. Written below this is the equation a superscript n equals a times a times ellipsis times a, implying an indeterminate number of “a”s being multiplied. a bracket is drawn below the “a”s being multiplied and labeled “n factors”.], alt: "a is shown with a superscripted n to the right of it. an arrow is drawn to a and labeled “base” while another arrow is drawn to the superscripted n and labeled “exponent”. Written below this is the equation a superscript n equals a times a times ellipsis times a, implying an indeterminate number of “a”s being multiplied. a bracket is drawn below the “a”s being multiplied and labeled “n factors”.", caption: none) The expression #math.equation(block: false, alt: "a to the power n")[$a^(n)$] is read #emph[a] to the #math.equation(block: false, alt: "n to the power t h")[$n^(t h)$] power. ] While we read #math.equation(block: false, alt: "a to the power n")[$a^(n)$] as “#emph[a] to the #math.equation(block: false, alt: "n to the power t h")[$n^(t h)$] power,” we usually read: - #math.equation(block: false, alt: "a squared")[$a^(2)$] “#emph[a] squared” - #math.equation(block: false, alt: "a cubed")[$a^(3)$] “#emph[a] cubed” We’ll see later why #math.equation(block: false, alt: "a squared")[$a^(2)$] and #math.equation(block: false, alt: "a cubed")[$a^(3)$] have special names. shows how we read some expressions with exponents. #figure(table( columns: 2, align: left, inset: 6pt, table.header([Expression], [In Words]), [#math.equation(block: false, alt: "7 squared")[$7^(2)$]], [7 to the second power or 7 squared], [#math.equation(block: false, alt: "5 cubed")[$5^(3)$]], [5 to the third power or 5 cubed], [#math.equation(block: false, alt: "9 to the power 4")[$9^(4)$]], [9 to the fourth power], [#math.equation(block: false, alt: "12 to the power 5")[$12^(5)$]], [#math.equation(block: false, alt: "12")[$12$] to the fifth power], )) #examplebox("Example 3")[][ Simplify: #math.equation(block: false, alt: "3 to the power 4 .")[$3^(4) .$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "3 to the power 4")[$3^(4)$]]), [Expand the expression.], [#math.equation(block: false, alt: "3 times 3 times 3 times 3")[$3 · 3 · 3 · 3$]], [Multiply left to right.], [#math.equation(block: false, alt: "9 times 3 times 3")[$9 · 3 · 3$]], [Multiply.], [#math.equation(block: false, alt: "27 times 3")[$27 · 3$]], [Multiply.], [81], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: ⓐ #math.equation(block: false, alt: "5 cubed")[$5^(3)$] ⓑ #math.equation(block: false, alt: "1 to the power 7 .")[$1^(7) .$] #solutionbox[ ⓐ 125 ⓑ 1 ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: ⓐ #math.equation(block: false, alt: "7 squared")[$7^(2)$] ⓑ #math.equation(block: false, alt: "0 to the power 5 .")[$0^(5) .$] #solutionbox[ ⓐ 49 ⓑ 0 ] ] === Simplify Expressions Using the Order of Operations To #strong[simplify an expression] means to do all the math possible. For example, to simplify #math.equation(block: false, alt: "4 times 2 plus 1")[$4 · 2 + 1$] we’d first multiply #math.equation(block: false, alt: "4 times 2")[$4 · 2$] to get 8 and then add the 1 to get 9. A good habit to develop is to work down the page, writing each step of the process below the previous step. The example just described would look like this: #math.equation(block: true, alt: "4 times 2 plus 1; 8 plus 1; 9")[$4 · 2 + 1 \ 8 + 1 \ 9$]By not using an #strong[equal sign] when you simplify an expression, you may avoid confusing expressions with equations. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Simplify an Expression] To #strong[simplify an expression], do all operations in the expression. ] We’ve introduced most of the symbols and notation used in algebra, but now we need to clarify the order of operations. Otherwise, expressions may have different meanings, and they may result in different values. For example, consider the expression: #math.equation(block: true, alt: "4 plus 3 times 7")[$4 + 3 · 7$]If you simplify this expression, what do you get? Some students say 49, #math.equation(block: true, alt: "4 plus 3 times 7; Since 4 plus 3 gives 7 ., 7 times 7; And 7 times 7 is 49 ., 49")[$& & & & & #h(0.8em) 4 + 3 · 7 \ "Since" #h(0.2em) 4 + 3 #h(0.2em) "gives" #h(0.2em) 7 . & & & & & #h(0.8em) 7 · 7 \ "And" #h(0.2em) 7 · 7 #h(0.2em) "is" #h(0.2em) 49 . & & & & & #h(0.8em) 49$]Others say 25, #math.equation(block: true, alt: "4 plus 3 times 7; Since 3 times 7 is 21 ., 4 plus 21; And 21 plus 4 makes 25 ., 25")[$& & & & & 4 + 3 · 7 \ "Since" #h(0.2em) 3 · 7 #h(0.2em) "is" #h(0.2em) 21 . & & & & & 4 + 21 \ "And" #h(0.2em) 21 + 4 #h(0.2em) "makes" #h(0.2em) 25 . & & & & & 25$]Imagine the confusion in our banking system if every problem had several different correct answers! The same expression should give the same result. So mathematicians early on established some guidelines that are called the #strong[Order of Operations]. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Perform the Order of Operations.] + Parentheses and Other Grouping Symbols - Simplify all expressions inside the parentheses or other grouping symbols, working on the innermost parentheses first. + Exponents - Simplify all expressions with exponents. + Multiplication and Division - Perform all multiplication and division in order from left to right. These operations have equal priority. + Addition and #strong[S]ubtraction - Perform all addition and subtraction in order from left to right. These operations have equal priority. ] Students often ask, “How will I remember the order?” Here is a way to help you remember: Take the first letter of each key word and substitute the silly phrase: “Please Excuse My Dear Aunt Sally.” #math.equation(block: true, alt: "P arentheses, P lease; E xponents, E xcuse; M ultiplication D ivision, M y D ear; A ddition S ubtraction, A unt S ally")[$"P" "arentheses" & & & "P" "lease" \ "E" "xponents" & & & "E" "xcuse" \ "M" "ultiplication" #h(0.2em) "D" "ivision" & & & "M" "y" #h(0.2em) "D" "ear" \ "A" "ddition" #h(0.2em) "S" "ubtraction" & & & "A" "unt" #h(0.2em) "S" "ally"$]It’s good that “#strong[M]y #strong[D]ear” goes together, as this reminds us that #strong[m]ultiplication and #strong[d]ivision have equal priority. We do not always do multiplication before division or always do division before multiplication. We do them in order from left to right. Similarly, “#strong[A]unt #strong[S]ally” goes together and so reminds us that #strong[a]ddition and #strong[s]ubtraction also have equal priority and we do them in order from left to right. Let’s try an example. #examplebox("Example 4")[][ Simplify: ⓐ #math.equation(block: false, alt: "4 plus 3 times 7")[$4 + 3 · 7$] ⓑ #math.equation(block: false, alt: "open parenthesis 4 plus 3 close parenthesis times 7 .")[$( 4 + 3 ) · 7 .$] #solutionbox[ ⓐ #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[A mathematical expression displays the equation 4+3⋅7 in black text on a white background.], alt: "A mathematical expression displays the equation 4+3⋅7 in black text on a white background.", caption: none)]), [Are there any #strong[p]arentheses? No.], [], [Are there any #strong[e]xponents? No.], [], [Is there any #strong[m]ultiplication or #strong[d]ivision? Yes.], [], [Multiply first.], [#figure(figph[A mathematical expression '4+3⋅7' is displayed on a white background. The number 4 and the plus sign are black, while the number 3, the multiplication dot, and the number 7 are red.], alt: "A mathematical expression '4+3⋅7' is displayed on a white background. The number 4 and the plus sign are black, while the number 3, the multiplication dot, and the number 7 are red.", caption: none)], [Add.], [#figure(figph[The image displays a simple mathematical equation, '4+21' in a black font against a white background.], alt: "The image displays a simple mathematical equation, '4+21' in a black font against a white background.", caption: none)], [], [#figure(figph[The number 25 is displayed in a dark gray font against a plain white background.], alt: "The number 25 is displayed in a dark gray font against a plain white background.", caption: none)], )) ⓑ #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[A mathematical expression shows the sum of 4 and 3, enclosed in parentheses, multiplied by 7. The expression is written as (4 + 3) ', 7.], alt: "A mathematical expression shows the sum of 4 and 3, enclosed in parentheses, multiplied by 7. The expression is written as (4 + 3) ', 7.", caption: none)]), [Are there any #strong[p]arentheses? Yes.], [#figure(figph[A mathematical expression showing (4+3) multiplied by 7. The numbers 4 and 3 are in red, while the plus sign, parentheses, multiplication dot, and the number 7 are in black or dark gray.], alt: "A mathematical expression showing (4+3) multiplied by 7. The numbers 4 and 3 are in red, while the plus sign, parentheses, multiplication dot, and the number 7 are in black or dark gray.", caption: none)], [Simplify inside the parentheses.], [#figure(figph[The number 7 is shown twice, with the first 7 enclosed in parentheses and colored red, followed by a black 7.], alt: "The number 7 is shown twice, with the first 7 enclosed in parentheses and colored red, followed by a black 7.", caption: none)], [Are there any #strong[e]xponents? No.], [], [Is there any #strong[m]ultiplication or #strong[d]ivision? Yes.], [], [Multiply.], [#figure(figph[The number '49' is prominently displayed against a plain white background, rendered in a dark gray or black font with a slight blur effect, suggesting it might be part of a larger digital display or a simple graphic.], alt: "The number '49' is prominently displayed against a plain white background, rendered in a dark gray or black font with a slight blur effect, suggesting it might be part of a larger digital display or a simple graphic.", caption: none)], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: ⓐ #math.equation(block: false, alt: "12 minus 5 times 2")[$12 − 5 · 2$] ⓑ #math.equation(block: false, alt: "open parenthesis 12 minus 5 close parenthesis times 2 .")[$( 12 − 5 ) · 2 .$] #solutionbox[ ⓐ 2 ⓑ 14 ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: ⓐ #math.equation(block: false, alt: "8 plus 3 times 9")[$8 + 3 · 9$] ⓑ #math.equation(block: false, alt: "open parenthesis 8 plus 3 close parenthesis times 9 .")[$( 8 + 3 ) · 9 .$] #solutionbox[ ⓐ 35 ⓑ 99 ] ] #examplebox("Example 5")[][ Simplify: #math.equation(block: false, alt: "18 divided by 6 plus 4 open parenthesis 5 minus 2 close parenthesis .")[$18 ÷ 6 + 4 ( 5 − 2 ) .$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([Parentheses? Yes, subtract first.], [#math.equation(block: false, alt: "18 divided by 6 plus 4 open parenthesis 5 minus 2 close parenthesis")[$18 ÷ 6 + 4 ( 5 − 2 )$] #linebreak() $#h(0.2em)$#figure(figph[A mathematical expression showing 18 divided by 6, plus 4 multiplied by 3, which is 18 ÷ 6 + 4(3).], alt: "A mathematical expression showing 18 divided by 6, plus 4 multiplied by 3, which is 18 ÷ 6 + 4(3).", caption: none)]), [Exponents? No.], [], [Multiplication or division? Yes.], [#figure(figph[A mathematical expression 18 ÷ 6 + 4(3) is displayed in red font on a white background, demonstrating the order of operations in arithmetic.], alt: "A mathematical expression 18 ÷ 6 + 4(3) is displayed in red font on a white background, demonstrating the order of operations in arithmetic.", caption: none)], [Divide first because we multiply and divide left to right.], [#figure(figph[A mathematical expression '3 + 4(3)' is displayed, where the '4(3)' part is highlighted in red, indicating a multiplication operation to be performed before addition according to order of operations.], alt: "A mathematical expression '3 + 4(3)' is displayed, where the '4(3)' part is highlighted in red, indicating a multiplication operation to be performed before addition according to order of operations.", caption: none)], [Any other multiplication or division? Yes.], [], [Multiply.], [#figure(figph[A simple mathematical equation is displayed, showing '3 + 12' in a clear, dark font against a white background, representing an addition problem.], alt: "A simple mathematical equation is displayed, showing '3 + 12' in a clear, dark font against a white background, representing an addition problem.", caption: none)], [Any other multiplication or division? No.], [], [Any addition or subtraction? Yes.], [#figure(figph[The number '15' is displayed in dark gray on a white background.], alt: "The number '15' is displayed in dark gray on a white background.", caption: none)], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "30 divided by 5 plus 10 open parenthesis 3 minus 2 close parenthesis .")[$30 ÷ 5 + 10 ( 3 − 2 ) .$] #solutionbox[ 16 ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "70 divided by 10 plus 4 open parenthesis 6 minus 2 close parenthesis .")[$70 ÷ 10 + 4 ( 6 − 2 ) .$] #solutionbox[ 23 ] ] When there are multiple grouping symbols, we simplify the innermost parentheses first and work outward. #examplebox("Example 6")[][ Simplify: #math.equation(block: false, alt: "5 plus 2 cubed plus 3 [ 6 minus 3 open parenthesis 4 minus 2 close parenthesis ] .")[$5 + 2^(3) + 3 [ 6 − 3 ( 4 − 2 ) ] .$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[A mathematical expression is displayed, featuring numbers and operations including addition, subtraction, exponentiation, and multiplication, organized with parentheses and brackets: 5 + 2^3 + 3\[6 - 3(4 - 2)\].], alt: "A mathematical expression is displayed, featuring numbers and operations including addition, subtraction, exponentiation, and multiplication, organized with parentheses and brackets: 5 + 2^3 + 3[6 - 3(4 - 2)].", caption: none)]), [Are there any parentheses (or other grouping symbol)? Yes.], [], [Focus on the parentheses that are inside the brackets.], [#figure(figph[A mathematical problem displaying an arithmetic expression: 5 + 2^3 + 3\[6 - 3(4 - 2)\]. The subtraction within the innermost parentheses, '4 - 2', is highlighted in red.], alt: "A mathematical problem displaying an arithmetic expression: 5 + 2^3 + 3[6 - 3(4 - 2)]. The subtraction within the innermost parentheses, '4 - 2', is highlighted in red.", caption: none)], [Subtract.], [#figure(figph[A mathematical expression reads 5 + 2^3 + 3\[6 - 3(2)\]. The '3(2)' part is highlighted in red, indicating a specific focus or step in solving the problem.], alt: "A mathematical expression reads 5 + 2^3 + 3[6 - 3(2)]. The '3(2)' part is highlighted in red, indicating a specific focus or step in solving the problem.", caption: none)], [Continue inside the brackets and multiply.], [#figure(figph[A mathematical expression displays 5 + 2 cubed + 3 multiplied by the quantity 6 minus 6, with the second 6 highlighted in red.], alt: "A mathematical expression displays 5 + 2 cubed + 3 multiplied by the quantity 6 minus 6, with the second 6 highlighted in red.", caption: none)], [Continue inside the brackets and subtract.], [#figure(figph[A mathematical expression showing '5 + 2^3 + 3\[0\]'. The number 0 is highlighted in red within the brackets, indicating a specific element or value.], alt: "A mathematical expression showing '5 + 2^3 + 3[0]'. The number 0 is highlighted in red within the brackets, indicating a specific element or value.", caption: none)], [The expression inside the brackets requires no further simplification.], [], [Are there any exponents? Yes.], [#figure(figph[A mathematical expression showing 5 plus 2 raised to the power of 3, plus 3 multiplied by 0.], alt: "A mathematical expression showing 5 plus 2 raised to the power of 3, plus 3 multiplied by 0.", caption: none)], [Simplify exponents.], [#figure(figph[A mathematical expression '5 + 8 + 3\[0\]' is displayed on a white background, with the '3\[0\]' part highlighted in red text, suggesting a specific focus on that segment of the equation.], alt: "A mathematical expression '5 + 8 + 3[0]' is displayed on a white background, with the '3[0]' part highlighted in red text, suggesting a specific focus on that segment of the equation.", caption: none)], [Is there any multiplication or division? Yes.], [], [Multiply.], [#figure(figph[An image showing the mathematical expression 5+8+0. The numbers '5' and '8' along with the first plus sign appear in red, while the second plus sign and the number '0' are black.], alt: "An image showing the mathematical expression 5+8+0. The numbers '5' and '8' along with the first plus sign appear in red, while the second plus sign and the number '0' are black.", caption: none)], [Is there any addition or subtraction? Yes.], [], [Add.], [#figure(figph[The mathematical expression '13 + 0' is displayed in a red, slightly blurred font against a white background.], alt: "The mathematical expression '13 + 0' is displayed in a red, slightly blurred font against a white background.", caption: none)], [Add.], [#figure(figph[13], alt: "13", caption: none)], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "9 plus 5 cubed minus [ 4 open parenthesis 9 plus 3 close parenthesis ] .")[$9 + 5^(3) − [ 4 ( 9 + 3 ) ] .$] #solutionbox[ 86 ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "7 squared minus 2 [ 4 open parenthesis 5 plus 1 close parenthesis ] .")[$7^(2) − 2 [ 4 ( 5 + 1 ) ] .$] #solutionbox[ 1 ] ] === Evaluate an Expression In the last few examples, we simplified expressions using the order of operations. Now we’ll evaluate some expressions—again following the order of operations. To #strong[evaluate an expression] means to find the value of the expression when the variable is replaced by a given number. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Evaluate an Expression] To #strong[evaluate an expression] means to find the value of the expression when the variable is replaced by a given number. ] To evaluate an expression, substitute that number for the variable in the expression and then simplify the expression. #examplebox("Example 7")[][ Evaluate #math.equation(block: false, alt: "7 x minus 4 ,")[$7 x − 4 ,$] when ⓐ #math.equation(block: false, alt: "x equals 5")[$x = 5$] and ⓑ #math.equation(block: false, alt: "x equals 1 .")[$x = 1 .$] #solutionbox[ ⓐ #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([#figure(figph[The text 'when x = 5' is displayed against a white background, with the number 5 subtly highlighted in red.], alt: "The text 'when x = 5' is displayed against a white background, with the number 5 subtly highlighted in red.", caption: none)], [#figure(figph[A mathematical expression '7x-4' is shown on a white background, featuring the number 7, the variable x, a minus sign, and the number 4.], alt: "A mathematical expression '7x-4' is shown on a white background, featuring the number 7, the variable x, a minus sign, and the number 4.", caption: none)]), [], [#figure(figph[A mathematical expression showing 7 multiplied by 5, with 4 subtracted from the result, appearing as 7(5)-4. The number 5 is highlighted in red.], alt: "A mathematical expression showing 7 multiplied by 5, with 4 subtracted from the result, appearing as 7(5)-4. The number 5 is highlighted in red.", caption: none)], [Multiply.], [#figure(figph[The image displays the numbers '35-4' in a dark gray font against a white background.], alt: "The image displays the numbers '35-4' in a dark gray font against a white background.", caption: none)], [Subtract.], [#figure(figph[The number '31' is clearly visible in the top right corner of a plain white background.], alt: "The number '31' is clearly visible in the top right corner of a plain white background.", caption: none)], )) ⓑ #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([#figure(figph[The text 'when x = 1' is displayed on a white background, with '1' highlighted in red.], alt: "The text 'when x = 1' is displayed on a white background, with '1' highlighted in red.", caption: none)], [#figure(figph[The mathematical expression '7x-4' is displayed on a white background, featuring the number 7, the variable x, a minus sign, and the number 4.], alt: "The mathematical expression '7x-4' is displayed on a white background, featuring the number 7, the variable x, a minus sign, and the number 4.", caption: none)]), [], [#figure(figph[A mathematical expression reads '7(1)-4', with the number 1 prominently highlighted in red.], alt: "A mathematical expression reads '7(1)-4', with the number 1 prominently highlighted in red.", caption: none)], [Multiply.], [#figure(figph[The numbers 7-4 are displayed on a white background, suggesting a score, date, or simple numerical notation.], alt: "The numbers 7-4 are displayed on a white background, suggesting a score, date, or simple numerical notation.", caption: none)], [Subtract.], [#figure(figph[A close-up of a large, dark grey numeral 3 on a clean white background.], alt: "A close-up of a large, dark grey numeral 3 on a clean white background.", caption: none)], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Evaluate #math.equation(block: false, alt: "8 x minus 3 ,")[$8 x − 3 ,$] when ⓐ #math.equation(block: false, alt: "x equals 2")[$x = 2$] and ⓑ #math.equation(block: false, alt: "x equals 1 .")[$x = 1 .$] #solutionbox[ ⓐ 13 ⓑ 5 ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Evaluate #math.equation(block: false, alt: "4 y minus 4 ,")[$4 y − 4 ,$] when ⓐ #math.equation(block: false, alt: "y equals 3")[$y = 3$] and ⓑ #math.equation(block: false, alt: "y equals 5 .")[$y = 5 .$] #solutionbox[ ⓐ 8 ⓑ 16 ] ] #examplebox("Example 8")[][ Evaluate the following for #math.equation(block: false, alt: "x equals 4 ,")[$x = 4 ,$] when ⓐ #math.equation(block: false, alt: "x squared")[$x^(2)$] ⓑ #math.equation(block: false, alt: "3 to the power x .")[$3^(x) .$] #solutionbox[ ⓐ #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "x squared")[$x^(2)$]]), [#figure(figph[The image displays the text 'Replace x with 4.' in a gray sans-serif font, with the number 4 highlighted in red, on a plain white background.], alt: "The image displays the text 'Replace x with 4.' in a gray sans-serif font, with the number 4 highlighted in red, on a plain white background.", caption: none)], [#figure(figph[A close-up image showing a large red number 4 raised to the power of 2.], alt: "A close-up image showing a large red number 4 raised to the power of 2.", caption: none)], [Use definition of exponent.], [#math.equation(block: false, alt: "4 times 4")[$4 · 4$]], [Simplify.], [#math.equation(block: false, alt: "16")[$16$]], )) ⓑ #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "3 to the power x")[$3^("x")$]]), [#figure(figph[The text 'Replace x with 4.' is displayed in a digital format. The word 'Replace' and the letters 'x with' are in a dark gray font, while the number '4' is highlighted in a red-orange color, followed by a dark gray period. The background is white.], alt: "The text 'Replace x with 4.' is displayed in a digital format. The word 'Replace' and the letters 'x with' are in a dark gray font, while the number '4' is highlighted in a red-orange color, followed by a dark gray period. The background is white.", caption: none)], [#figure(figph[The mathematical expression showing 3 raised to the power of x, written as 3^x, where 3 is the base and x is the exponent.], alt: "The mathematical expression showing 3 raised to the power of x, written as 3^x, where 3 is the base and x is the exponent.", caption: none)], [Use definition of exponent.], [#math.equation(block: false, alt: "3 times 3 times 3 times 3")[$3 · 3 · 3 · 3$]], [Simplify.], [#math.equation(block: false, alt: "81")[$81$]], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Evaluate #math.equation(block: false, alt: "x equals 3 ,")[$x = 3 ,$] when ⓐ #math.equation(block: false, alt: "x squared")[$x^(2)$] ⓑ #math.equation(block: false, alt: "4 to the power x .")[$4^(x) .$] #solutionbox[ ⓐ #math.equation(block: false, alt: "9")[$9$] ⓑ #math.equation(block: false, alt: "64")[$64$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Evaluate #math.equation(block: false, alt: "x equals 6 ,")[$x = 6 ,$] when ⓐ #math.equation(block: false, alt: "x cubed")[$x^(3)$] ⓑ #math.equation(block: false, alt: "2 to the power x .")[$2^(x) .$] #solutionbox[ ⓐ #math.equation(block: false, alt: "216")[$216$] ⓑ #math.equation(block: false, alt: "64")[$64$] ] ] #examplebox("Example 9")[][ Evaluate #math.equation(block: false, alt: "2 x squared plus 3 x plus 8")[$2 x^(2) + 3 x + 8$] when #math.equation(block: false, alt: "x equals 4 .")[$x = 4 .$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "2 x squared plus 3 x plus 8")[$2 x^(2) + 3 x + 8$]]), [#figure(figph[The text 'Substitute x = 4.' is shown, with the number 4 highlighted in red.], alt: "The text 'Substitute x = 4.' is shown, with the number 4 highlighted in red.", caption: none)], [#figure(figph[A mathematical expression reads 2x squared plus 3x plus 8, in a black font on a white background.], alt: "A mathematical expression reads 2x squared plus 3x plus 8, in a black font on a white background.", caption: none)], [Follow the order of operations.], [#math.equation(block: false, alt: "2 open parenthesis 16 close parenthesis plus 3 open parenthesis 4 close parenthesis plus 8")[$2 ( 16 ) + 3 ( 4 ) + 8$]], [], [#math.equation(block: false, alt: "32 plus 12 plus 8")[$32 + 12 + 8$]], [], [#math.equation(block: false, alt: "52")[$52$]], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Evaluate #math.equation(block: false, alt: "3 x squared plus 4 x plus 1")[$3 x^(2) + 4 x + 1$] when #math.equation(block: false, alt: "x equals 3 .")[$x = 3 .$] #solutionbox[ 40 ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Evaluate #math.equation(block: false, alt: "6 x squared minus 4 x minus 7")[$6 x^(2) − 4 x − 7$] when #math.equation(block: false, alt: "x equals 2 .")[$x = 2 .$] #solutionbox[ 9 ] ] === Identify and Combine Like Terms Algebraic expressions are made up of terms. A #strong[term] is a constant, or the product of a constant and one or more variables. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Term] A #strong[term] is a constant, or the product of a constant and one or more variables. ] Examples of terms are #math.equation(block: false, alt: "7 , y , 5 x squared , 9 a , and b to the power 5 .")[$7 , y , 5 x^(2) , 9 a , "and" #h(0.2em) b^(5) .$] The constant that multiplies the variable is called the #strong[coefficient]. #linebreak() #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Coefficient] The #strong[coefficient] of a term is the constant that multiplies the variable in a term. ] Think of the coefficient as the number in front of the variable. The coefficient of the term 3#emph[x] is 3. When we write #emph[x], the coefficient is 1, since #math.equation(block: false, alt: "x equals 1 times x .")[$x = 1 · x .$] #examplebox("Example 10")[][ Identify the coefficient of each term: ⓐ 14#emph[y] ⓑ #math.equation(block: false, alt: "15 x squared")[$15 x^(2)$] ⓒ #emph[a]. #solutionbox[ ⓐ The coefficient of 14#emph[y] is 14. #linebreak() #linebreak() ⓑ The coefficient of #math.equation(block: false, alt: "15 x squared")[$15 x^(2)$] is 15. #linebreak() #linebreak() ⓒ The coefficient of #emph[a] is 1 since #math.equation(block: false, alt: "a equals 1 a .")[$a = 1 #h(0.2em) a .$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Identify the coefficient of each term: ⓐ #math.equation(block: false, alt: "17 x")[$17 x$] ⓑ #math.equation(block: false, alt: "41 b squared")[$41 b^(2)$] ⓒ #emph[z]. #solutionbox[ ⓐ 17 ⓑ 41 ⓒ 1 ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Identify the coefficient of each term: ⓐ 9#emph[p] ⓑ #math.equation(block: false, alt: "13 a cubed")[$13 a^(3)$] ⓒ #math.equation(block: false, alt: "y cubed .")[$y^(3) .$] #solutionbox[ ⓐ 9 ⓑ 13 ⓒ 1 ] ] Some terms share common traits. Look at the following 6 terms. Which ones seem to have traits in common? #math.equation(block: true, alt: "5 x, 7, n squared, 4, 3 x, 9 n squared")[$5 x & & & 7 & & & n^(2) & & & 4 & & & 3 x & & & 9 n^(2)$]The 7 and the 4 are both constant terms. The #emph[5x] and the 3#emph[x] are both terms with #emph[x]. The #math.equation(block: false, alt: "n squared")[$n^(2)$] and the #math.equation(block: false, alt: "9 n squared")[$9 n^(2)$] are both terms with #math.equation(block: false, alt: "n squared .")[$n^(2) .$] When two terms are constants or have the same variable and exponent, we say they are #strong[like terms]. - 7 and 4 are like terms. - 5#emph[x] and 3#emph[x] are like terms. - #math.equation(block: false, alt: "x squared")[$x^(2)$] and #math.equation(block: false, alt: "9 x squared")[$9 x^(2)$] are like terms. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Like Terms] Terms that are either constants or have the same variables raised to the same powers are called #strong[like terms]. ] #examplebox("Example 11")[][ Identify the like terms: #math.equation(block: false, alt: "y cubed ,")[$y^(3) ,$] #math.equation(block: false, alt: "7 x squared ,")[$7 x^(2) ,$] 14, 23, #math.equation(block: false, alt: "4 y cubed ,")[$4 y^(3) ,$] 9#emph[x], #math.equation(block: false, alt: "5 x squared .")[$5 x^(2) .$] #solutionbox[ #math.equation(block: false, alt: "y cubed")[$y^(3)$] and #math.equation(block: false, alt: "4 y cubed")[$4 y^(3)$] are like terms because both have #math.equation(block: false, alt: "y cubed ;")[$y^(3) ;$] the variable and the exponent match. #math.equation(block: false, alt: "7 x squared")[$7 x^(2)$] and #math.equation(block: false, alt: "5 x squared")[$5 x^(2)$] are like terms because both have #math.equation(block: false, alt: "x squared ;")[$x^(2) ;$] the variable and the exponent match. 14 and 23 are like terms because both are constants. There is no other term like 9#emph[x]. ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Identify the like terms: #math.equation(block: false, alt: "9 ,")[$9 ,$] #math.equation(block: false, alt: "2 x cubed ,")[$2 x^(3) ,$] #math.equation(block: false, alt: "y squared ,")[$y^(2) ,$] #math.equation(block: false, alt: "8 x cubed ,")[$8 x^(3) ,$] #math.equation(block: false, alt: "15 ,")[$15 ,$] #math.equation(block: false, alt: "9 y ,")[$9 y ,$] #math.equation(block: false, alt: "11 y squared .")[$11 y^(2) .$] #solutionbox[ 9 and 15, #math.equation(block: false, alt: "y squared")[$y^(2)$] and #math.equation(block: false, alt: "11 y squared ,")[$11 y^(2) ,$] #math.equation(block: false, alt: "2 x cubed")[$2 x^(3)$] and #math.equation(block: false, alt: "8 x cubed")[$8 x^(3)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Identify the like terms: #math.equation(block: false, alt: "4 x cubed ,")[$4 x^(3) ,$] #math.equation(block: false, alt: "8 x squared ,")[$8 x^(2) ,$] 19, #math.equation(block: false, alt: "3 x squared ,")[$3 x^(2) ,$] 24, #math.equation(block: false, alt: "6 x cubed .")[$6 x^(3) .$] #solutionbox[ 19 and 24, #math.equation(block: false, alt: "8 x squared")[$8 x^(2)$] and #math.equation(block: false, alt: "3 x squared ,")[$3 x^(2) ,$] #math.equation(block: false, alt: "4 x cubed")[$4 x^(3)$] and #math.equation(block: false, alt: "6 x cubed")[$6 x^(3)$] ] ] Adding or subtracting terms forms an expression. In the expression #math.equation(block: false, alt: "2 x squared plus 3 x plus 8 ,")[$2 x^(2) + 3 x + 8 ,$] from , the three terms are #math.equation(block: false, alt: "2 x squared , 3 x ,")[$2 x^(2) , 3 x ,$] and 8. #examplebox("Example 12")[][ Identify the terms in each expression. + ⓐ #math.equation(block: false, alt: "9 x squared plus 7 x plus 12")[$9 x^(2) + 7 x + 12$] + ⓑ #math.equation(block: false, alt: "8 x plus 3 y")[$8 x + 3 y$] #solutionbox[ + ⓐ The terms of #math.equation(block: false, alt: "9 x squared plus 7 x plus 12")[$9 x^(2) + 7 x + 12$] are #math.equation(block: false, alt: "9 x squared ,")[$9 x^(2) ,$] 7#emph[x], and 12. #linebreak() #linebreak() + ⓑ The terms of #math.equation(block: false, alt: "8 x plus 3 y")[$8 x + 3 y$] are 8#emph[x] and 3#emph[y]. ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Identify the terms in the expression #math.equation(block: false, alt: "4 x squared plus 5 x plus 17 .")[$4 x^(2) + 5 x + 17 .$] #solutionbox[ #math.equation(block: true, alt: "4 x squared , 5 x , 17")[$4 x^(2) , 5 x , 17$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Identify the terms in the expression #math.equation(block: false, alt: "5 x plus 2 y .")[$5 x + 2 y .$] #solutionbox[ 5#emph[x], 2#emph[y] ] ] If there are like terms in an expression, you can simplify the expression by combining the like terms. What do you think #math.equation(block: false, alt: "4 x plus 7 x plus x")[$4 x + 7 x + x$] would simplify to? If you thought 12#emph[x], you would be right! #math.equation(block: true, alt: "4 x plus 7 x plus x; x plus x plus x plus x plus x plus x plus x plus x plus x plus x plus x plus x; 12 x")[$4 x + 7 x + x \ x + x + x + x #h(1em) + x + x + x + x + x + x + x #h(1em) + x \ 12 x$]Add the coefficients and keep the same variable. It doesn’t matter what #emph[x] is—if you have 4 of something and add 7 more of the same thing and then add 1 more, the result is 12 of them. For example, 4 oranges plus 7 oranges plus 1 orange is 12 oranges. We will discuss the mathematical properties behind this later. Simplify: #math.equation(block: false, alt: "4 x plus 7 x plus x .")[$4 x + 7 x + x .$] Add the coefficients. 12#emph[x] #examplebox("Example 13")[How To Combine Like Terms][ Simplify: #math.equation(block: false, alt: "2 x squared plus 3 x plus 7 plus x squared plus 4 x plus 5 .")[$2 x^(2) + 3 x + 7 + x^(2) + 4 x + 5 .$] #solutionbox[ #figure(figph[Three lines of instructions are listed in a column on the left side of the image while four algebraic expressions are listed on the right. The first line of instruction on the left says: “Step 1. Identify like terms.” Across from step 1 in the right column is the algebraic expression: 2x squared plus 3x plus 7 plus x squared plus 4x plus 5. One line down on the right, the same algebraic expression is repeated, except each of the terms appears in one of three colors to illustrate that these are like terms: 2x squared and x squared appear as red, illustrating that these are like terms; 3x and 4x appear as blue, illustrating that these are also like terms; 7 and 5 appear as green, illustrating that these are like terms as well.], alt: "Three lines of instructions are listed in a column on the left side of the image while four algebraic expressions are listed on the right. The first line of instruction on the left says: “Step 1. Identify like terms.” Across from step 1 in the right column is the algebraic expression: 2x squared plus 3x plus 7 plus x squared plus 4x plus 5. One line down on the right, the same algebraic expression is repeated, except each of the terms appears in one of three colors to illustrate that these are like terms: 2x squared and x squared appear as red, illustrating that these are like terms; 3x and 4x appear as blue, illustrating that these are also like terms; 7 and 5 appear as green, illustrating that these are like terms as well.", caption: none) #figure(figph[The second line of instruction on the left says: “Step 2. Rearrange the expression so the like terms are together. Across from step 2 in the right column is the original algebraic expression with terms reordered so that like terms appear side by side: 2x squared plus x2, both written in red, plus 3x plus 4x, both written n blue, plus 7 plus 5, both written in green.], alt: "The second line of instruction on the left says: “Step 2. Rearrange the expression so the like terms are together. Across from step 2 in the right column is the original algebraic expression with terms reordered so that like terms appear side by side: 2x squared plus x2, both written in red, plus 3x plus 4x, both written n blue, plus 7 plus 5, both written in green.", caption: none) #figure(figph[The third line of instruction on the left says: “Step 3. Combine like terms.” Across from step 3 in the right column is the algebraic expression with like terms combined: 3x squared in red, plus 7x in blue, plus 12 in green.], alt: "The third line of instruction on the left says: “Step 3. Combine like terms.” Across from step 3 in the right column is the algebraic expression with like terms combined: 3x squared in red, plus 7x in blue, plus 12 in green.", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "3 x squared plus 7 x plus 9 plus 7 x squared plus 9 x plus 8 .")[$3 x^(2) + 7 x + 9 + 7 x^(2) + 9 x + 8 .$] #solutionbox[ #math.equation(block: true, alt: "10 x squared plus 16 x plus 17")[$10 x^(2) + 16 x + 17$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "4 y squared plus 5 y plus 2 plus 8 y squared plus 4 y plus 5 .")[$4 y^(2) + 5 y + 2 + 8 y^(2) + 4 y + 5 .$] #solutionbox[ #math.equation(block: true, alt: "12 y squared plus 9 y plus 7")[$12 y^(2) + 9 y + 7$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Combine Like Terms.] + Identify like terms. + Rearrange the expression so like terms are together. + Add or subtract the coefficients and keep the same variable for each group of like terms. ] === Translate an English Phrase to an Algebraic Expression In the last section, we listed many operation symbols that are used in algebra, then we translated expressions and equations into English phrases and sentences. Now we’ll reverse the process. We’ll translate English phrases into algebraic expressions. The symbols and variables we’ve talked about will help us do that. summarizes them. #figure(table( columns: 3, align: left, inset: 6pt, table.header([Operation], [Phrase], [Expression]), [Addition], [#emph[a] plus #emph[b] #linebreak() the sum of #math.equation(block: false, alt: "a")[$a$] and #emph[b] #linebreak() #math.equation(block: false, alt: "a")[$a$] increased by #emph[b] #linebreak() #emph[b] more than #math.equation(block: false, alt: "a")[$a$] #linebreak() the total of #math.equation(block: false, alt: "a")[$a$] and #emph[b] #linebreak() #emph[b] added to #math.equation(block: false, alt: "a")[$a$]], [#math.equation(block: false, alt: "a plus b")[$a + b$]], [Subtraction], [#math.equation(block: false, alt: "a")[$a$] minus #emph[b] #linebreak() the difference of #math.equation(block: false, alt: "a")[$a$] and #emph[b] #linebreak() #math.equation(block: false, alt: "a")[$a$] decreased by #emph[b] #linebreak() #emph[b] less than #math.equation(block: false, alt: "a")[$a$] #linebreak() #emph[b] subtracted from #math.equation(block: false, alt: "a")[$a$]], [#math.equation(block: false, alt: "a minus b")[$a − b$]], [Multiplication], [#math.equation(block: false, alt: "a")[$a$] times #emph[b] #linebreak() the product of #math.equation(block: false, alt: "a")[$a$] and #emph[b] #linebreak() twice #math.equation(block: false, alt: "a")[$a$]], [#math.equation(block: false, alt: "a times b , a b , a open parenthesis b close parenthesis , open parenthesis a close parenthesis open parenthesis b close parenthesis")[$a · b , a b , a ( b ) , ( a ) ( b )$] #linebreak() #linebreak() 2#emph[a]], [Division], [#math.equation(block: false, alt: "a")[$a$] divided by #emph[b] #linebreak() the quotient of #math.equation(block: false, alt: "a")[$a$] and #emph[b] #linebreak() the ratio of #math.equation(block: false, alt: "a")[$a$] and #emph[b] #linebreak() #emph[b] divided into #math.equation(block: false, alt: "a")[$a$]], [#math.equation(block: false, alt: "a divided by b , a / b , the fraction a over b , b a")[$a ÷ b , a "/" b , frac(a, b) , b ") " overline(a)$]], )) Look closely at these phrases using the four operations: #figure(figph[Four phrases are shown. The first reads “the sum of a and b”, where the words “of” and “and” are written in red. The second reads “the difference of a and b”, where the words “of” and “and” are written in red. The third reads “the product of a and b”, where the words “of” and “and” are written in red. The fourth reads “the quotient of a and b”, where the words “of” and “and” are written in red.], alt: "Four phrases are shown. The first reads “the sum of a and b”, where the words “of” and “and” are written in red. The second reads “the difference of a and b”, where the words “of” and “and” are written in red. The third reads “the product of a and b”, where the words “of” and “and” are written in red. The fourth reads “the quotient of a and b”, where the words “of” and “and” are written in red.", caption: none) Each phrase tells us to operate on two numbers. Look for the words #emph[of] and #emph[and] to find the numbers. #examplebox("Example 14")[][ Translate each English phrase into an algebraic expression: ⓐ the difference of #math.equation(block: false, alt: "17 x")[$17 x$] and #math.equation(block: false, alt: "5")[$5$] ⓑ the quotient of #math.equation(block: false, alt: "10 x squared")[$10 x^(2)$] and #math.equation(block: false, alt: "7 .")[$7 .$] #solutionbox[ + ⓐ The key word is #emph[difference], which tells us the operation is subtraction. Look for the words #emph[of] and #emph[and t]o find the numbers to subtract. #linebreak() #figure(figph[The phrase “the difference of 17x and 5”, where the words “of” and “and” are written in red, is written above the phrase “17 x minus 5”. a final phrase written below reads “17 x, minus sign, 5”.], alt: "The phrase “the difference of 17x and 5”, where the words “of” and “and” are written in red, is written above the phrase “17 x minus 5”. a final phrase written below reads “17 x, minus sign, 5”.", caption: none) #linebreak() #linebreak() + ⓑ The key word is “quotient,” which tells us the operation is division. #figure(figph[The phrase “the quotient of 10x squared and 7”, where the words “of” and “and” are written in red, is written above the expression “divide 10x squared by 7”. The expression written below reads “10x squared, division sign,v7”.], alt: "The phrase “the quotient of 10x squared and 7”, where the words “of” and “and” are written in red, is written above the expression “divide 10x squared by 7”. The expression written below reads “10x squared, division sign,v7”.", caption: none) This can also be written #math.equation(block: false, alt: "10 x squared / 7 or the fraction 10 x squared over 7 .")[$10 x^(2) "/" 7 #h(0.2em) "or" #h(0.2em) frac(10 x^(2), 7) .$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Translate the English phrase into an algebraic expression: ⓐ the difference of #math.equation(block: false, alt: "14 x squared")[$14 x^(2)$] and 13 ⓑ the quotient of 12#emph[x] and 2. #solutionbox[ ⓐ #math.equation(block: false, alt: "14 x squared minus 13")[$14 x^(2) − 13$] ⓑ #math.equation(block: false, alt: "12 x divided by 2")[$12 x ÷ 2$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Translate the English phrase into an algebraic expression: ⓐ the sum of #math.equation(block: false, alt: "17 y squared")[$17 y^(2)$] and #math.equation(block: false, alt: "19")[$19$] ⓑ the product of #math.equation(block: false, alt: "7")[$7$] and #math.equation(block: false, alt: "y .")[$y .$] #solutionbox[ ⓐ #math.equation(block: false, alt: "17 y squared plus 19")[$17 y^(2) + 19$] ⓑ 7#emph[y] ] ] How old will you be in eight years? What age is eight more years than your age now? Did you add 8 to your present age? Eight “more than” means 8 added to your present age. How old were you seven years ago? This is 7 years less than your age now. You subtract 7 from your present age. Seven “less than” means 7 subtracted from your present age. #examplebox("Example 15")[][ Translate the English phrase into an algebraic expression: ⓐ Seventeen more than #emph[y] ⓑ Nine less than #math.equation(block: false, alt: "9 x squared .")[$9 x^(2) .$] #solutionbox[ + ⓐ The key words are #emph[more than.] They tell us the operation is addition. #emph[More than] means “added to.” #linebreak() #math.equation(block: true, alt: "Seventeen more than y; Seventeen added to y; y plus 17")[$"Seventeen more than" #h(0.2em) y \ "Seventeen added to" #h(0.2em) y \ y + 17$] + ⓑ The key words are #emph[less than]. They tell us to subtract. #emph[Less than] means “subtracted from.” #linebreak() #math.equation(block: true, alt: "Nine less than 9 x squared; Nine subtracted from 9 x squared; 9 x squared minus 9")[$#h(5em) "Nine less than" #h(0.2em) 9 x^(2) \ #h(5em) "Nine subtracted from" #h(0.2em) 9 x^(2) \ #h(5em) 9 x^(2) − 9$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Translate the English phrase into an algebraic expression: ⓐ Eleven more than #math.equation(block: false, alt: "x")[$x$] ⓑ Fourteen less than #math.equation(block: false, alt: "11 a .")[$11 a .$] #solutionbox[ ⓐ #math.equation(block: false, alt: "x plus 11")[$x + 11$] ⓑ #math.equation(block: false, alt: "11 a minus 14")[$11 a − 14$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Translate the English phrase into an algebraic expression: ⓐ 13 more than #emph[z] ⓑ 18 less than 8#emph[x]. #solutionbox[ ⓐ #math.equation(block: false, alt: "z plus 13")[$z + 13$] ⓑ #math.equation(block: false, alt: "8 x minus 18")[$8 x − 18$] ] ] #examplebox("Example 16")[][ Translate the English phrase into an algebraic expression: ⓐ five times the sum of #emph[m] and #emph[n] ⓑ the sum of five times #emph[m] and #emph[n]. #solutionbox[ There are two operation words—#emph[times] tells us to multiply and #emph[sum] tells us to add. + ⓐ Because we are multiplying 5 times the sum we need parentheses around the sum of #emph[m] and #emph[n], #math.equation(block: false, alt: "open parenthesis m plus n close parenthesis .")[$( m + n ) .$] This forces us to determine the sum first. (Remember the order of operations.) #linebreak() #math.equation(block: true, alt: "five times the sum of m and n; 5 open parenthesis m plus n close parenthesis")[$"five times the sum of" #h(0.2em) m #h(0.2em) "and" #h(0.2em) n \ 5 #h(0.2em) ( m + n )$] + ⓑ To take a sum, we look for the words “of” and “and” to see what is being added. Here we are taking the sum #emph[of] five times #emph[m] and #emph[n]. #linebreak() #math.equation(block: true, alt: "the sum of five times m and n; 5 m plus n")[$"the sum" #h(0.2em) "of" #h(0.2em) "five times" #h(0.2em) m #h(0.2em) "and" #h(0.2em) n \ 5 m + n$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Translate the English phrase into an algebraic expression: ⓐ four times the sum of #emph[p] and #emph[q] ⓑ the sum of four times #emph[p] and #emph[q]. #solutionbox[ ⓐ #math.equation(block: false, alt: "4 open parenthesis p plus q close parenthesis")[$4 ( p + q )$] ⓑ #math.equation(block: false, alt: "4 p plus q")[$4 p + q$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Translate the English phrase into an algebraic expression: ⓐ the difference of two times #emph[x] and 8, ⓑ two times the difference of #emph[x] and 8. #solutionbox[ ⓐ #math.equation(block: false, alt: "2 x minus 8")[$2 x − 8$] ⓑ #math.equation(block: false, alt: "2 open parenthesis x minus 8 close parenthesis")[$2 ( x − 8 )$] ] ] Later in this course, we’ll apply our skills in algebra to solving applications. The first step will be to translate an English phrase to an algebraic expression. We’ll see how to do this in the next two examples. #examplebox("Example 17")[][ The width of a rectangle is 6 less than the length. Let #emph[l] represent the length of the rectangle. Write an expression for the width of the rectangle. #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([Write a phrase about the width of the rectangle.], [6 less than the length]), [Substitute #math.equation(block: false, alt: "l")[$l$] for "the length."], [6 less than #math.equation(block: false, alt: "l")[$l$]], [Rewrite "less than" as "subtracted from."], [6 subtracted from #math.equation(block: false, alt: "l")[$l$]], [Translate the phrase into algebra.], [#math.equation(block: false, alt: "l minus 6")[$l − 6$]], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ The length of a rectangle is 7 less than the width. Let #emph[w] represent the width of the rectangle. Write an expression for the length of the rectangle. #solutionbox[ #math.equation(block: true, alt: "w minus 7")[$w − 7$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ The width of a rectangle is 6 less than the length. Let #emph[l] represent the length of the rectangle. Write an expression for the width of the rectangle. #solutionbox[ #math.equation(block: true, alt: "l minus 6")[$l − 6$] ] ] #examplebox("Example 18")[][ June has dimes and quarters in her purse. The number of dimes is three less than four times the number of quarters. Let #emph[q] represent the number of quarters. Write an expression for the number of dimes. #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([Write a phrase about the number of dimes.], [three less than four times the number of quarters]), [Substitute #math.equation(block: false, alt: "q")[$q$] for the number of quarters.], [3 less than 4 times #math.equation(block: false, alt: "q")[$q$]], [Translate "4 times #math.equation(block: false, alt: "q")[$q$]."], [3 less than #math.equation(block: false, alt: "4 q")[$4 q$]], [Translate the phase into algebra.], [#math.equation(block: false, alt: "4 q minus 3")[$4 q − 3$]], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Geoffrey has dimes and quarters in his pocket. The number of dimes is eight less than four times the number of quarters. Let #emph[q] represent the number of quarters. Write an expression for the number of dimes. #solutionbox[ #math.equation(block: true, alt: "4 q minus 8")[$4 q − 8$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Lauren has dimes and nickels in her purse. The number of dimes is three more than seven times the number of nickels. Let #emph[n] represent the number of nickels. Write an expression for the number of dimes. #solutionbox[ #math.equation(block: true, alt: "7 n plus 3")[$7 n + 3$] ] ] === Key Concepts - #strong[Notation                      The result is…] #linebreak() #linebreak() #math.equation(block: false, alt: "∘ a plus b, the sum of a and b; ∘ a minus b, the difference of a and b; ∘ a times b , a b , open parenthesis a close parenthesis open parenthesis b close parenthesis open parenthesis a close parenthesis b , a open parenthesis b close parenthesis, the product of a and b; ∘ a divided by b , a / b , the fraction a over b , b a, the quotient of a and b")[$#h(1.8em) ∘ #h(0.6em) a + b & & & #h(3em) "the sum of" #h(0.2em) a #h(0.2em) "and" #h(0.2em) b \ #h(1.8em) ∘ #h(0.6em) a − b & & & #h(3em) "the difference of" #h(0.2em) a #h(0.2em) "and" #h(0.2em) b \ #h(1.8em) ∘ #h(0.6em) a · b , a b , ( a ) ( b ) #h(0.2em) ( a ) b , a ( b ) & & & #h(3em) "the product of" #h(0.2em) a #h(0.2em) "and" #h(0.2em) b \ #h(1.8em) ∘ #h(0.6em) a ÷ b , a "/" b , frac(a, b) , b ") " overline(a) & & & #h(3em) "the quotient of" #h(0.2em) a #h(0.2em) "and" #h(0.2em) b$] - #strong[Inequality] #linebreak() #linebreak() #math.equation(block: false, alt: "∘ a less than b is read “ a is less than b ”, a is to the left of b on the number line; ∘ a greater than b is read “ a is greater than b ”, a is to the right of b on the number line")[$#h(1.8em) ∘ #h(0.6em) a < b #h(0.2em) "is read" #h(0.2em) "“" a #h(0.2em) "is less than" #h(0.2em) b "”" & & & a #h(0.2em) "is to the left of" #h(0.2em) b #h(0.2em) "on the number line" \ #h(1.8em) ∘ #h(0.6em) a > b #h(0.2em) "is read" #h(0.2em) "“" a #h(0.2em) "is greater than" #h(0.2em) b "”" & & & a #h(0.2em) "is to the right of" #h(0.2em) b #h(0.2em) "on the number line"$] - #strong[Inequality Symbols                 Words] #linebreak() #linebreak() #math.equation(block: false, alt: "∘ a not equal to b, a is not equal to b; ∘ a less than b, a is less than b; ∘ a less than or equal to b, a is less than or equal to b; ∘ a greater than b, a is greater than b; ∘ a greater than or equal to b, a is greater than or equal to b")[$#h(1.8em) ∘ #h(0.6em) a ≠ b & & & #h(11em) a #h(0.2em) "is" #h(0.2em) "not equal to" #h(0.2em) b \ #h(1.8em) ∘ #h(0.6em) a < b & & & #h(11em) a #h(0.2em) "is" #h(0.2em) "less than" #h(0.2em) b \ #h(1.8em) ∘ #h(0.6em) a ≤ b & & & #h(11em) a #h(0.2em) "is" #h(0.2em) "less than or equal to" #h(0.2em) b \ #h(1.8em) ∘ #h(0.6em) a > b & & & #h(11em) a #h(0.2em) "is" #h(0.2em) "greater than" #h(0.2em) b \ #h(1.8em) ∘ #h(0.6em) a ≥ b & & & #h(11em) a #h(0.2em) "is" #h(0.2em) "greater than or equal to" #h(0.2em) b$] - #strong[Grouping Symbols] - Parentheses #math.equation(block: false, alt: "open parenthesis close parenthesis")[$( #h(0.2em) )$] - Brackets #math.equation(block: false, alt: "[ ]")[$[ #h(0.2em) ]$] - Braces #math.equation(block: false, alt: "{ }")[$\{ #h(0.2em) \}$] - #strong[Exponential Notation] - #math.equation(block: false, alt: "a to the power n")[$a^(n)$] means multiply #math.equation(block: false, alt: "a")[$a$] by itself, #math.equation(block: false, alt: "n")[$n$] times. The expression #math.equation(block: false, alt: "a to the power n")[$a^(n)$] is read #math.equation(block: false, alt: "a")[$a$] to the #math.equation(block: false, alt: "n to the power t h")[$n^(t h)$] power. - #strong[Order of Operations:] When simplifying mathematical expressions perform the operations in the following order: + Parentheses and other Grouping Symbols: Simplify all expressions inside the parentheses or other grouping symbols, working on the innermost parentheses first. + Exponents: Simplify all expressions with exponents. + Multiplication and Division: Perform all multiplication and division in order from left to right. These operations have equal priority. + Addition and Subtraction: Perform all addition and subtraction in order from left to right. These operations have equal priority. - #strong[Combine Like Terms] + Identify like terms. + Rearrange the expression so like terms are together. + Add or subtract the coefficients and keep the same variable for each group of like terms. ==== Practice Makes Perfect #strong[Use Variables and Algebraic Symbols] In the following exercises, translate from algebra to English. #math.equation(block: true, alt: "16 minus 9")[$16 − 9$] #solutionbox[ 16 minus 9, the difference of sixteen and nine ] #math.equation(block: true, alt: "3 times 9")[$3 · 9$] #math.equation(block: true, alt: "28 divided by 4")[$28 ÷ 4$] #solutionbox[ 28 divided by 4, the quotient of twenty-eight and four ] #math.equation(block: true, alt: "x plus 11")[$x + 11$] #math.equation(block: true, alt: "open parenthesis 2 close parenthesis open parenthesis 7 close parenthesis")[$( 2 ) ( 7 )$] #solutionbox[ 2 times 7, the product of two and seven ] #math.equation(block: true, alt: "open parenthesis 4 close parenthesis open parenthesis 8 close parenthesis")[$( 4 ) ( 8 )$] #math.equation(block: true, alt: "14 less than 21")[$14 < 21$] #solutionbox[ fourteen is less than twenty-one ] #math.equation(block: true, alt: "17 less than 35")[$17 < 35$] #math.equation(block: true, alt: "36 greater than or equal to 19")[$36 ≥ 19$] #solutionbox[ thirty-six is greater than or equal to nineteen ] #math.equation(block: true, alt: "6 n equals 36")[$6 n = 36$] #math.equation(block: true, alt: "y minus 1 greater than 6")[$y − 1 > 6$] #solutionbox[ #emph[y] minus 1 is greater than 6, the difference of #emph[y] and one is greater than six ] #math.equation(block: true, alt: "y minus 4 greater than 8")[$y − 4 > 8$] #math.equation(block: true, alt: "2 less than or equal to 18 divided by 6")[$2 ≤ 18 ÷ 6$] #solutionbox[ 2 is less than or equal to 18 divided by 6; 2 is less than or equal to the quotient of eighteen and six ] #math.equation(block: true, alt: "a not equal to 1 times 12")[$a ≠ 1 · 12$] In the following exercises, determine if each is an expression or an equation. #math.equation(block: true, alt: "9 times 6 equals 54")[$9 · 6 = 54$] #solutionbox[ equation ] #math.equation(block: true, alt: "7 times 9 equals 63")[$7 · 9 = 63$] #math.equation(block: true, alt: "5 times 4 plus 3")[$5 · 4 + 3$] #solutionbox[ expression ] #math.equation(block: true, alt: "x plus 7")[$x + 7$] #math.equation(block: true, alt: "x plus 9")[$x + 9$] #solutionbox[ expression ] #math.equation(block: true, alt: "y minus 5 equals 25")[$y − 5 = 25$] #strong[Simplify Expressions Using the Order of Operations] In the following exercises, simplify each expression. #math.equation(block: true, alt: "5 cubed")[$5^(3)$] #solutionbox[ 125 ] #math.equation(block: true, alt: "8 cubed")[$8^(3)$] #math.equation(block: true, alt: "2 to the power 8")[$2^(8)$] #solutionbox[ 256 ] #math.equation(block: true, alt: "10 to the power 5")[$10^(5)$] In the following exercises, simplify using the order of operations. ⓐ #math.equation(block: false, alt: "3 plus 8 times 5")[$3 + 8 · 5$] ⓑ #math.equation(block: false, alt: "open parenthesis 3 plus 8 close parenthesis times 5")[$( 3 + 8 ) · 5$] #solutionbox[ ⓐ 43 ⓑ 55 ] ⓐ #math.equation(block: false, alt: "2 plus 6 times 3")[$2 + 6 · 3$] ⓑ #math.equation(block: false, alt: "open parenthesis 2 plus 6 close parenthesis times 3")[$( 2 + 6 ) · 3$] #math.equation(block: true, alt: "2 cubed minus 12 divided by open parenthesis 9 minus 5 close parenthesis")[$2^(3) − 12 ÷ ( 9 − 5 )$] #solutionbox[ 5 ] #math.equation(block: true, alt: "3 squared minus 18 divided by open parenthesis 11 minus 5 close parenthesis")[$3^(2) − 18 ÷ ( 11 − 5 )$] #math.equation(block: true, alt: "3 times 8 plus 5 times 2")[$3 · 8 + 5 · 2$] #solutionbox[ 34 ] #math.equation(block: true, alt: "4 times 7 plus 3 times 5")[$4 · 7 + 3 · 5$] #math.equation(block: true, alt: "2 plus 8 open parenthesis 6 plus 1 close parenthesis")[$2 + 8 ( 6 + 1 )$] #solutionbox[ 58 ] #math.equation(block: true, alt: "4 plus 6 open parenthesis 3 plus 6 close parenthesis")[$4 + 6 ( 3 + 6 )$] #math.equation(block: true, alt: "4 times 12 / 8")[$4 · 12 "/" 8$] #solutionbox[ 6 ] #math.equation(block: true, alt: "2 times 36 / 6")[$2 · 36 "/" 6$] #math.equation(block: true, alt: "open parenthesis 6 plus 10 close parenthesis divided by open parenthesis 2 plus 2 close parenthesis")[$( 6 + 10 ) ÷ ( 2 + 2 )$] #solutionbox[ 4 ] #math.equation(block: true, alt: "open parenthesis 9 plus 12 close parenthesis divided by open parenthesis 3 plus 4 close parenthesis")[$( 9 + 12 ) ÷ ( 3 + 4 )$] #math.equation(block: true, alt: "20 divided by 4 plus 6 times 5")[$20 ÷ 4 + 6 · 5$] #solutionbox[ 35 ] #math.equation(block: true, alt: "33 divided by 3 plus 8 times 2")[$33 ÷ 3 + 8 · 2$] #math.equation(block: true, alt: "3 squared plus 7 squared")[$3^(2) + 7^(2)$] #solutionbox[ 58 ] #math.equation(block: true, alt: "open parenthesis 3 plus 7 close parenthesis squared")[$attach(( 3 + 7 ), t: 2)$] #math.equation(block: true, alt: "3 open parenthesis 1 plus 9 times 6 close parenthesis minus 4 squared")[$3 ( 1 + 9 · 6 ) − 4^(2)$] #solutionbox[ 149 ] #math.equation(block: true, alt: "5 open parenthesis 2 plus 8 times 4 close parenthesis minus 7 squared")[$5 ( 2 + 8 · 4 ) − 7^(2)$] #math.equation(block: true, alt: "2 [ 1 plus 3 open parenthesis 10 minus 2 close parenthesis ]")[$2 [ 1 + 3 ( 10 − 2 ) ]$] #solutionbox[ 50 ] #math.equation(block: true, alt: "5 [ 2 plus 4 open parenthesis 3 minus 2 close parenthesis ]")[$5 [ 2 + 4 ( 3 − 2 ) ]$] #strong[Evaluate an Expression] In the following exercises, evaluate the following expressions. #math.equation(block: false, alt: "7 x plus 8")[$7 x + 8$] when #math.equation(block: false, alt: "x equals 2")[$x = 2$] #solutionbox[ 22 ] #math.equation(block: false, alt: "8 x minus 6")[$8 x − 6$] when #math.equation(block: false, alt: "x equals 7")[$x = 7$] #math.equation(block: false, alt: "x squared")[$x^(2)$] when #math.equation(block: false, alt: "x equals 12")[$x = 12$] #solutionbox[ 144 ] #math.equation(block: false, alt: "x cubed")[$x^(3)$] when #math.equation(block: false, alt: "x equals 5")[$x = 5$] #math.equation(block: false, alt: "x to the power 5")[$x^(5)$] when #math.equation(block: false, alt: "x equals 2")[$x = 2$] #solutionbox[ 32 ] #math.equation(block: false, alt: "4 to the power x")[$4^(x)$] when #math.equation(block: false, alt: "x equals 2")[$x = 2$] #math.equation(block: false, alt: "x squared plus 3 x minus 7")[$x^(2) + 3 x − 7$] when #math.equation(block: false, alt: "x equals 4")[$x = 4$] #solutionbox[ 21 ] #math.equation(block: false, alt: "6 x plus 3 y minus 9")[$6 x + 3 y − 9$] when #linebreak() #math.equation(block: false, alt: "x equals 6 , y equals 9")[$x = 6 , y = 9$] #math.equation(block: false, alt: "open parenthesis x minus y close parenthesis squared")[$attach(( x − y ), t: 2)$] when #linebreak() #math.equation(block: false, alt: "x equals 10 , y equals 7")[$x = 10 , y = 7$] #solutionbox[ 9 ] #math.equation(block: false, alt: "open parenthesis x plus y close parenthesis squared")[$attach(( x + y ), t: 2)$] when #math.equation(block: false, alt: "x equals 6 , y equals 9")[$x = 6 , y = 9$] #math.equation(block: false, alt: "a squared plus b squared")[$a^(2) + b^(2)$] when #math.equation(block: false, alt: "a equals 3 , b equals 8")[$a = 3 , b = 8$] #solutionbox[ 73 ] #math.equation(block: false, alt: "r squared minus s squared")[$r^(2) − s^(2)$] when #math.equation(block: false, alt: "r equals 12 , s equals 5")[$r = 12 , s = 5$] #math.equation(block: false, alt: "2 l plus 2 w")[$2 l + 2 w$] when #linebreak() #math.equation(block: false, alt: "l equals 15 , w equals 12")[$l = 15 , w = 12$] #solutionbox[ 54 ] #math.equation(block: false, alt: "2 l plus 2 w")[$2 l + 2 w$] when #linebreak() #math.equation(block: false, alt: "l equals 18 , w equals 14")[$l = 18 , w = 14$] #strong[Simplify Expressions by Combining Like Terms] In the following exercises, identify the coefficient of each term. 8#emph[a] #solutionbox[ 8 ] 13#emph[m] #math.equation(block: true, alt: "5 r squared")[$5 r^(2)$] #solutionbox[ 5 ] #math.equation(block: true, alt: "6 x cubed")[$6 x^(3)$] In the following exercises, identify the like terms. #math.equation(block: true, alt: "x cubed , 8 x , 14 , 8 y , 5 , 8 x cubed")[$x^(3) , 8 x , 14 , 8 y , 5 , 8 x^(3)$] #solutionbox[ #math.equation(block: true, alt: "x cubed and 8 x cubed , 14 and 5")[$x^(3) #h(0.2em) "and" #h(0.2em) 8 x^(3) , 14 #h(0.2em) "and" #h(0.2em) 5$] ] #math.equation(block: true, alt: "6 z , 3 w squared , 1 , 6 z squared , 4 z , w squared")[$6 z , 3 w^(2) , 1 , 6 z^(2) , 4 z , w^(2)$] #math.equation(block: true, alt: "9 a , a squared , 16 , 16 b squared , 4 , 9 b squared")[$9 a , a^(2) , 16 , 16 b^(2) , 4 , 9 b^(2)$] #solutionbox[ #math.equation(block: true, alt: "16 and 4 , 16 b squared and 9 b squared")[$16 #h(0.2em) "and" #h(0.2em) 4 , 16 b^(2) #h(0.2em) "and" #h(0.2em) 9 b^(2)$] ] #math.equation(block: true, alt: "3 , 25 r squared , 10 s , 10 r , 4 r squared , 3 s")[$3 , 25 r^(2) , 10 s , 10 r , 4 r^(2) , 3 s$] In the following exercises, identify the terms in each expression. #math.equation(block: true, alt: "15 x squared plus 6 x plus 2")[$15 x^(2) + 6 x + 2$] #solutionbox[ #math.equation(block: true, alt: "15 x squared , 6 x , 2")[$15 x^(2) , 6 x , 2$] ] #math.equation(block: true, alt: "11 x squared plus 8 x plus 5")[$11 x^(2) + 8 x + 5$] #math.equation(block: true, alt: "10 y cubed plus y plus 2")[$10 y^(3) + y + 2$] #solutionbox[ #math.equation(block: true, alt: "10 y cubed , y , 2")[$10 y^(3) , y , 2$] ] #math.equation(block: true, alt: "9 y cubed plus y plus 5")[$9 y^(3) + y + 5$] In the following exercises, simplify the following expressions by combining like terms. #math.equation(block: true, alt: "10 x plus 3 x")[$10 x + 3 x$] #solutionbox[ 13#emph[x] ] #math.equation(block: true, alt: "15 x plus 4 x")[$15 x + 4 x$] #math.equation(block: true, alt: "4 c plus 2 c plus c")[$4 c + 2 c + c$] #solutionbox[ 7#emph[c] ] #math.equation(block: true, alt: "6 y plus 4 y plus y")[$6 y + 4 y + y$] #math.equation(block: true, alt: "7 u plus 2 plus 3 u plus 1")[$7 u + 2 + 3 u + 1$] #solutionbox[ #math.equation(block: true, alt: "10 u plus 3")[$10 u + 3$] ] #math.equation(block: true, alt: "8 d plus 6 plus 2 d plus 5")[$8 d + 6 + 2 d + 5$] #math.equation(block: true, alt: "10 a plus 7 plus 5 a minus 2 plus 7 a minus 4")[$10 a + 7 + 5 a − 2 + 7 a − 4$] #solutionbox[ #math.equation(block: true, alt: "22 a plus 1")[$22 a + 1$] ] #math.equation(block: true, alt: "7 c plus 4 plus 6 c minus 3 plus 9 c minus 1")[$7 c + 4 + 6 c − 3 + 9 c − 1$] #math.equation(block: true, alt: "3 x squared plus 12 x plus 11 plus 14 x squared plus 8 x plus 5")[$3 x^(2) + 12 x + 11 + 14 x^(2) + 8 x + 5$] #solutionbox[ #math.equation(block: true, alt: "17 x squared plus 20 x plus 16")[$17 x^(2) + 20 x + 16$] ] #math.equation(block: true, alt: "5 b squared plus 9 b plus 10 plus 2 b squared plus 3 b minus 4")[$5 b^(2) + 9 b + 10 + 2 b^(2) + 3 b − 4$] #strong[Translate an English Phrase to an Algebraic Expression] In the following exercises, translate the phrases into algebraic expressions. the difference of 14 and 9 #solutionbox[ #math.equation(block: true, alt: "14 minus 9")[$14 − 9$] ] the difference of 19 and 8 the product of 9 and 7 #solutionbox[ #math.equation(block: true, alt: "9 times 7")[$9 · 7$] ] the product of 8 and 7 the quotient of 36 and 9 #solutionbox[ #math.equation(block: true, alt: "36 divided by 9")[$36 ÷ 9$] ] the quotient of 42 and 7 the sum of #emph[8x] and 3#emph[x] #solutionbox[ #math.equation(block: true, alt: "8 x plus 3 x")[$8 x + 3 x$] ] the sum of #emph[13x] and 3#emph[x] the quotient of #emph[y] and 3 #solutionbox[ #math.equation(block: true, alt: "the fraction y over 3")[$frac(y, 3)$] ] the quotient of #emph[y] and 8 eight times the difference of #emph[y] and nine #solutionbox[ #math.equation(block: true, alt: "8 open parenthesis y minus 9 close parenthesis")[$8 ( y − 9 )$] ] seven times the difference of #emph[y] and one Eric has rock and classical CDs in his car. The number of rock CDs is 3 more than the number of classical CDs. Let #emph[c] represent the number of classical CDs. Write an expression for the number of rock CDs. #solutionbox[ #math.equation(block: true, alt: "c plus 3")[$c + 3$] ] The number of girls in a second-grade class is 4 less than the number of boys. Let #emph[b] represent the number of boys. Write an expression for the number of girls. Greg has nickels and pennies in his pocket. The number of pennies is seven less than twice the number of nickels. Let #math.equation(block: false, alt: "n")[$n$] represent the number of nickels. Write an expression for the number of pennies. #solutionbox[ #math.equation(block: true, alt: "2 n minus 7")[$2 n − 7$] ] Jeannette has \$5 and \$10 bills in her wallet. The number of fives is three more than six times the number of tens. Let #emph[t] represent the number of tens. Write an expression for the number of fives. ==== Everyday Math #strong[Car insurance] Justin’s car insurance has a \$750 deductible per incident. This means that he pays \$750 and his insurance company will pay all costs beyond \$750. If Justin files a claim for \$2,100. + ⓐ how much will he pay? + ⓑ how much will his insurance company pay? #solutionbox[ ⓐ \$750 ⓑ \$1,350 ] #strong[Home insurance] Armando’s home insurance has a \$2,500 deductible per incident. This means that he pays \$2,500 and the insurance company will pay all costs beyond \$2,500. If Armando files a claim for \$19,400. + ⓐ how much will he pay? + ⓑ how much will the insurance company pay? ==== Writing Exercises Explain the difference between an expression and an equation. #solutionbox[ Answers may vary ] Why is it important to use the order of operations to simplify an expression? Explain how you identify the like terms in the expression #math.equation(block: false, alt: "8 a squared plus 4 a plus 9 minus a squared minus 1 .")[$8 a^(2) + 4 a + 9 − a^(2) − 1 .$] #solutionbox[ Answers may vary ] Explain the difference between the phrases “4 times the sum of #emph[x] and #emph[y]” and “the sum of 4 times #emph[x] and #emph[y].” ==== Self Check ⓐ Use this checklist to evaluate your mastery of the objectives of this section. #figure(figph[A table is shown that is composed of four columns and six rows. The header row reads, from left to right, “I can …”, “Confidently”, “With some help” and “No – I don’t get it!”. The phrases in the first column read “use variables and algebraic symbols.”, “simplify expressions using the order of operations.”, “evaluate an expression.”, “identify and combine like terms.”, and “translate English phrases to algebraic expressions.”], alt: "A table is shown that is composed of four columns and six rows. The header row reads, from left to right, “I can …”, “Confidently”, “With some help” and “No – I don’t get it!”. The phrases in the first column read “use variables and algebraic symbols.”, “simplify expressions using the order of operations.”, “evaluate an expression.”, “identify and combine like terms.”, and “translate English phrases to algebraic expressions.”", caption: none) ⓑ After reviewing this checklist, what will you do to become confident for all objectives?