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== 2.1#h(0.6em)Use the Language of Algebra

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Before you get started, take this readiness quiz.

#math.equation(block: false, alt: "Add: 43 plus 69 .")[$"Add:" #h(0.2em) 43 + 69 .$] #linebreak() If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "112")[$112$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#math.equation(block: false, alt: "Multiply: open parenthesis 896 close parenthesis 201 .")[$"Multiply:" #h(0.2em) ( 896 ) 201 .$] #linebreak() If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "180 , 096")[$180 , 096$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#math.equation(block: false, alt: "Divide: 7,263 divided by 9 .")[$"Divide:" #h(0.2em) 7,263 ÷ 9 .$] #linebreak() If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "807")[$807$]
]
]


=== Use Variables and Algebraic Symbols

Greg and Alex have the same birthday, but they were born in different years. This year Greg is #math.equation(block: false, alt: "20")[$20$] years old and Alex is #math.equation(block: false, alt: "23 ,")[$23 ,$] so Alex is #math.equation(block: false, alt: "3")[$3$] years older than Greg. When Greg was #math.equation(block: false, alt: "12 ,")[$12 ,$] Alex was #math.equation(block: false, alt: "15 .")[$15 .$] When Greg is #math.equation(block: false, alt: "35 ,")[$35 ,$] Alex will be #math.equation(block: false, alt: "38 .")[$38 .$] No matter what Greg’s age is, Alex’s age will always be #math.equation(block: false, alt: "3")[$3$] years more, right?

In the language of algebra, we say that Greg’s age and Alex’s age are #strong[variable] and the three is a #strong[constant]. The ages change, or vary, so age is a variable. The #math.equation(block: false, alt: "3")[$3$] years between them always stays the same, so the age difference is the constant.

In algebra, letters of the alphabet are used to represent variables. Suppose we call Greg’s age #math.equation(block: false, alt: "g .")[$g .$] 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")[$g + 3$]],
))

Letters are used to represent variables. Letters often used for variables are #math.equation(block: false, alt: "x , y , a , b , and c .")[$x , y , a , b , "and" #h(0.2em) c .$]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Variables and Constants]

A variable is a letter that represents a number or quantity whose value may change.

A constant is a number whose value always stays the same.
]

To write algebraically, we need some symbols as well as numbers and variables. There are several types of symbols we will be using. In Whole Numbers, we introduced the symbols for the four basic arithmetic #strong[operations]: addition, subtraction, multiplication, and division. We will summarize them here, along with words we use for the operations and the result.

#figure(table(
  columns: 4, align: left, inset: 6pt,
  table.header([#strong[Operation]], [#strong[Notation]], [#strong[Say:]], [#strong[The result is…]]),
  [Addition], [#math.equation(block: false, alt: "a plus b")[$a + b$]], [#math.equation(block: false, alt: "a plus b")[$a #h(0.2em) "plus" #h(0.2em) b$]], [the sum of #math.equation(block: false, alt: "a")[$a$] and #math.equation(block: false, alt: "b")[$b$]],
  [Subtraction], [#math.equation(block: false, alt: "a minus b")[$a − b$]], [#math.equation(block: false, alt: "a minus b")[$a #h(0.2em) "minus" #h(0.2em) b$]], [the difference of #math.equation(block: false, alt: "a")[$a$] and #math.equation(block: false, alt: "b")[$b$]],
  [Multiplication], [#math.equation(block: false, alt: "a times b , open parenthesis a close parenthesis open parenthesis b close parenthesis , open parenthesis a close parenthesis b , a open parenthesis b close parenthesis")[$a · b , ( a ) ( b ) , ( a ) b , a ( b )$]], [#math.equation(block: false, alt: "a times b")[$a #h(0.2em) "times" #h(0.2em) b$]], [The product of #math.equation(block: false, alt: "a")[$a$] and #math.equation(block: false, alt: "b")[$b$]],
  [Division], [#math.equation(block: false, alt: "a divided by b , a / b , the fraction a over b , b a")[$a ÷ b , a / b , #h(0.2em) frac(a, b) , b ") " overline(a)$]], [#math.equation(block: false, alt: "a")[$a$] divided by #math.equation(block: false, alt: "b")[$b$]], [The quotient of #math.equation(block: false, alt: "a")[$a$] and #math.equation(block: false, alt: "b")[$b$]],
))

In algebra, the cross symbol, #math.equation(block: false, alt: "times ,")[$× ,$] is not used to show multiplication because that symbol may cause confusion. Does #math.equation(block: false, alt: "3 x y")[$3 x y$] mean #math.equation(block: false, alt: "3 times y")[$3 #h(0.2em) × #h(0.2em) y$] (three times #math.equation(block: false, alt: "y")[$y$]) or #math.equation(block: false, alt: "3 times x times y")[$3 · x · y$] (three times #math.equation(block: false, alt: "x times y")[$x #h(0.2em) "times" #h(0.2em) y$])? To make it clear, use ⋅ or parentheses for multiplication.

We perform these operations on two numbers. When translating from symbolic form to words, or from words to symbolic form, pay attention to the words #emph[of] or #emph[and] to help you find the numbers.

- The #emph[sum] #strong[#emph[of]] #math.equation(block: false, alt: "5")[$5$] #strong[#emph[and]] #math.equation(block: false, alt: "3")[$3$] means add #math.equation(block: false, alt: "5")[$5$] plus #math.equation(block: false, alt: "3 ,")[$3 ,$] which we write as #math.equation(block: false, alt: "5 plus 3 .")[$5 + 3 .$]
- The #emph[difference] #strong[#emph[of]] #math.equation(block: false, alt: "9")[$9$] #strong[#emph[and]] #math.equation(block: false, alt: "2")[$2$] means subtract #math.equation(block: false, alt: "9")[$9$] minus #math.equation(block: false, alt: "2 ,")[$2 ,$] which we write as #math.equation(block: false, alt: "9 minus 2 .")[$9 − 2 .$]
- The #emph[product] #strong[#emph[of]] #math.equation(block: false, alt: "4")[$4$] #strong[#emph[and]] #math.equation(block: false, alt: "8")[$8$] means multiply #math.equation(block: false, alt: "4")[$4$] times #math.equation(block: false, alt: "8 ,")[$8 ,$] which we can write as #math.equation(block: false, alt: "4 times 8 .")[$4 · 8 .$]
- The #emph[quotient] #strong[#emph[of]] #math.equation(block: false, alt: "20")[$20$] #strong[#emph[and]] #math.equation(block: false, alt: "5")[$5$] means divide #math.equation(block: false, alt: "20")[$20$] by #math.equation(block: false, alt: "5 ,")[$5 ,$] which we can write as #math.equation(block: false, alt: "20 divided by 5 .")[$20 ÷ 5 .$]

#examplebox("Example 1")[][
Translate from algebra to words:

+ ⓐ #math.equation(block: false, alt: "12 plus 14")[$#h(0.2em) 12 + 14 #h(0.2em)$]
+ ⓑ #math.equation(block: false, alt: "open parenthesis 30 close parenthesis open parenthesis 5 close parenthesis")[$#h(0.2em) ( 30 ) ( 5 ) #h(0.2em)$]
+ ⓒ #math.equation(block: false, alt: "64 divided by 8")[$#h(0.2em) 64 ÷ 8 #h(0.2em)$]
+ ⓓ #math.equation(block: false, alt: "x minus y")[$#h(0.2em) x − y$]

#solutionbox[
#figure(table(
  columns: 1, align: left, inset: 6pt,
  [ⓐ],
  [#math.equation(block: false, alt: "12 plus 14")[$12 + 14$]],
  [12 plus 14],
  [the sum of twelve and fourteen],
))

#figure(table(
  columns: 1, align: left, inset: 6pt,
  [ⓑ],
  [#math.equation(block: false, alt: "open parenthesis 30 close parenthesis open parenthesis 5 close parenthesis")[$( 30 ) ( 5 )$]],
  [30 times 5],
  [the product of thirty and five],
))

#figure(table(
  columns: 1, align: left, inset: 6pt,
  [ⓒ],
  [#math.equation(block: false, alt: "64 divided by 8")[$64 ÷ 8$]],
  [64 divided by 8],
  [the quotient of sixty-four and eight],
))

#figure(table(
  columns: 1, align: left, inset: 6pt,
  [ⓓ],
  [#math.equation(block: false, alt: "x minus y")[$x − y$]],
  [#math.equation(block: false, alt: "x")[$x$] minus #math.equation(block: false, alt: "y")[$y$]],
  [the difference of #math.equation(block: false, alt: "x")[$x$] and #math.equation(block: false, alt: "y")[$y$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate from algebra to words.

+ ⓐ #math.equation(block: false, alt: "18 plus 11")[$#h(0.2em) 18 + 11 #h(0.4em)$]
+ ⓑ #math.equation(block: false, alt: "open parenthesis 27 close parenthesis open parenthesis 9 close parenthesis")[$#h(0.2em) ( 27 ) ( 9 ) #h(0.4em)$]
+ ⓒ #math.equation(block: false, alt: "84 divided by 7")[$#h(0.2em) 84 ÷ 7$]
+ ⓓ #math.equation(block: false, alt: "p minus q")[$p − q$]

#solutionbox[
+ ⓐ 18 plus 11; the sum of eighteen and eleven
+ ⓑ 27 times 9; the product of twenty-seven and nine
+ ⓒ 84 divided by 7; the quotient of eighty-four and seven
+ ⓓ #emph[p] minus #emph[q]; the difference of #emph[p] and #emph[q]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate from algebra to words.

+ ⓐ #math.equation(block: false, alt: "47 minus 19")[$#h(0.2em) 47 − 19 #h(0.4em)$]
+ ⓑ #math.equation(block: false, alt: "72 divided by 9")[$#h(0.2em) 72 ÷ 9 #h(0.4em)$]
+ ⓒ #math.equation(block: false, alt: "m plus n")[$#h(0.2em) m + n #h(0.4em)$]
+ ⓓ #math.equation(block: false, alt: "open parenthesis 13 close parenthesis open parenthesis 7 close parenthesis")[$#h(0.2em) ( 13 ) ( 7 )$]

#solutionbox[
+ ⓐ 47 minus 19; the difference of forty-seven and nineteen
+ ⓑ 72 divided by 9; the quotient of seventy-two and nine
+ ⓒ #emph[m] plus #emph[n]; the sum of #emph[m] and #emph[n]
+ ⓓ 13 times 7; the product of thirteen and seven
]
]

When two quantities have the same value, we say they are equal and connect them with an #emph[equal sign].

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Equality Symbol]

#math.equation(block: true, alt: "a equals b is read a is equal to b")[$a = b #h(0.2em) "is read" #h(0.2em) a #h(0.2em) "is equal to" #h(0.2em) b$]

The symbol #math.equation(block: false, alt: "equals")[$=$] is called the #strong[equal sign].
]

An #strong[inequality] is used in algebra to compare two quantities that may have different values. The number line can help you understand inequalities. Remember that on the number line the numbers get larger as they go from left to right. So if we know that #math.equation(block: false, alt: "b")[$b$] is greater than #math.equation(block: false, alt: "a ,")[$a ,$] it means that #math.equation(block: false, alt: "b")[$b$] is to the right of #math.equation(block: false, alt: "a")[$a$] on the number line. We use the symbols #math.equation(block: false, alt: "“<”")[$"“<”"$] and #math.equation(block: false, alt: "“>”")[$"“>”"$] for inequalities.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Inequality]

#math.equation(block: false, alt: "a less than b")[$a < b$] is read #math.equation(block: false, alt: "a")[$a$] is less than #math.equation(block: false, alt: "b")[$b$]

#math.equation(block: false, alt: "a")[$a$] is to the left of #math.equation(block: false, alt: "b")[$b$] on the number line

#figure(figph[The figure shows a horizontal number line that begins with the letter a on the left then the letter b to its right.], alt: "The figure shows a horizontal number line that begins with the letter a on the left then the letter b to its right.", caption: none)

#math.equation(block: false, alt: "a greater than b")[$a > b$] is read #math.equation(block: false, alt: "a")[$a$] is greater than #math.equation(block: false, alt: "b")[$b$]

#math.equation(block: false, alt: "a")[$a$] is to the right of #math.equation(block: false, alt: "b")[$b$] on the number line

#figure(figph[The figure shows a horizontal number line that begins with the letter b on the left then the letter a to its right.], alt: "The figure shows a horizontal number line that begins with the letter b on the left then the letter a to its right.", caption: none)
]

The expressions #math.equation(block: false, alt: "a less than b and a greater than b")[$a < b #h(0.2em) "and" #h(0.2em) a > b$] can be read from left-to-right or right-to-left, though in English we usually read from left-to-right. In general,

#math.equation(block: true, alt: "a less than b is equivalent to b greater than a . For example, 7 less than 11 is equivalent to 11 greater than 7 .; a greater than b is equivalent to b less than a . For example, 17 greater than 4 is equivalent to 4 less than 17 .")[$a < b #h(0.2em) "is equivalent to" #h(0.2em) b > a . #h(2em) "For example," #h(0.2em) 7 < 11 #h(0.2em) "is equivalent to" #h(0.2em) 11 > 7 . \ a > b #h(0.2em) "is equivalent to" #h(0.2em) b < a . #h(2em) "For example," #h(0.2em) 17 > 4 #h(0.2em) "is equivalent to" #h(0.2em) 4 < 17 .$]When we write an inequality symbol with a line under it, such as #math.equation(block: false, alt: "a less than or equal to b ,")[$a ≤ b ,$] it means #math.equation(block: false, alt: "a less than b")[$a < b$] or #math.equation(block: false, alt: "a equals b .")[$a = b .$] We read this #math.equation(block: false, alt: "a")[$a$] is less than or equal to #math.equation(block: false, alt: "b .")[$b .$] Also, if we put a slash through an equal sign, #math.equation(block: false, alt: "≠,")[$"≠,"$] it means not equal.

We summarize the symbols of equality and inequality in .

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Algebraic Notation], [Say]),
  [#math.equation(block: false, alt: "a equals b")[$a = b$]], [#math.equation(block: false, alt: "a")[$a$] is equal to #math.equation(block: false, alt: "b")[$b$]],
  [#math.equation(block: false, alt: "a not equal to b")[$a ≠ b$]], [#math.equation(block: false, alt: "a")[$a$] is not equal to #math.equation(block: false, alt: "b")[$b$]],
  [#math.equation(block: false, alt: "a less than b")[$a < b$]], [#math.equation(block: false, alt: "a")[$a$] is less than #math.equation(block: false, alt: "b")[$b$]],
  [#math.equation(block: false, alt: "a greater than b")[$a > b$]], [#math.equation(block: false, alt: "a")[$a$] is greater than #math.equation(block: false, alt: "b")[$b$]],
  [#math.equation(block: false, alt: "a less than or equal to b")[$a ≤ b$]], [#math.equation(block: false, alt: "a")[$a$] is less than or equal to #math.equation(block: false, alt: "b")[$b$]],
  [#math.equation(block: false, alt: "a greater than or equal to b")[$a ≥ b$]], [#math.equation(block: false, alt: "a")[$a$] is greater than or equal to #math.equation(block: false, alt: "b")[$b$]],
))

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Symbols #math.equation(block: false, alt: "less than")[$<$] and #math.equation(block: false, alt: "greater than")[$>$]]

The symbols #math.equation(block: false, alt: "less than")[$<$] and #math.equation(block: false, alt: "greater than")[$>$] each have a smaller side and a larger side.

smaller side   #math.equation(block: false, alt: "less than")[$<$]      larger side   #linebreak() 
larger side     #math.equation(block: false, alt: "greater than")[$>$]      smaller side

The smaller side of the symbol faces the smaller number and the larger faces the larger number.
]

#examplebox("Example 2")[][
Translate from algebra to words:

+ ⓐ #math.equation(block: false, alt: "20 less than or equal to 35")[$#h(0.2em) 20 ≤ 35$]
+ ⓑ #math.equation(block: false, alt: "11 not equal to 15 minus 3")[$#h(0.2em) 11 ≠ 15 − 3$]
+ ⓒ #math.equation(block: false, alt: "9 greater than 10 divided by 2")[$#h(0.2em) 9 > 10 ÷ 2$]
+ ⓓ #math.equation(block: false, alt: "x plus 2 less than 10")[$#h(0.2em) x + 2 < 10$]

#solutionbox[
#figure(table(
  columns: 1, align: left, inset: 6pt,
  [ⓐ],
  [#math.equation(block: false, alt: "20 less than or equal to 35")[$20 ≤ 35$]],
  [20 is less than or equal to 35],
))

#figure(table(
  columns: 1, align: left, inset: 6pt,
  [ⓑ],
  [#math.equation(block: false, alt: "11 not equal to 15 minus 3")[$11 ≠ 15 − 3$]],
  [11 is not equal to 15 minus 3],
))

#figure(table(
  columns: 1, align: left, inset: 6pt,
  [ⓒ],
  [#math.equation(block: false, alt: "9 greater than 10 divided by 2")[$9 > 10 ÷ 2$]],
  [9 is greater than 10 divided by 2],
))

#figure(table(
  columns: 1, align: left, inset: 6pt,
  [ⓓ],
  [#math.equation(block: false, alt: "x plus 2 less than 10")[$x + 2 < 10$]],
  [#math.equation(block: false, alt: "x")[$x$] plus 2 is less than 10],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate from algebra to words.

+ ⓐ #math.equation(block: false, alt: "14 less than or equal to 27")[$#h(0.2em) 14 ≤ 27 #h(0.2em)$]
+ ⓑ #math.equation(block: false, alt: "19 minus 2 not equal to 8")[$#h(0.2em) 19 − 2 ≠ 8 #h(0.2em)$]
+ ⓒ #math.equation(block: false, alt: "12 greater than 4 divided by 2")[$#h(0.2em) 12 > 4 ÷ 2 #h(0.2em)$]
+ ⓓ #math.equation(block: false, alt: "x minus 7 less than 1")[$#h(0.2em) x − 7 < 1$]

#solutionbox[
+ ⓐ fourteen is less than or equal to twenty-seven
+ ⓑ nineteen minus two is not equal to eight
+ ⓒ twelve is greater than four divided by two
+ ⓓ #emph[x] minus seven is less than one
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate from algebra to words.

+ ⓐ #math.equation(block: false, alt: "19 greater than or equal to 15")[$#h(0.2em) 19 ≥ 15 #h(0.2em)$]
+ ⓑ #math.equation(block: false, alt: "7 equals 12 minus 5")[$#h(0.2em) 7 = 12 − 5 #h(0.2em)$]
+ ⓒ #math.equation(block: false, alt: "15 divided by 3 less than 8")[$#h(0.2em) 15 ÷ 3 < 8 #h(0.2em)$]
+ ⓓ #math.equation(block: false, alt: "y minus 3 greater than 6")[$#h(0.2em) y − 3 > 6$]

#solutionbox[
+ ⓐ nineteen is greater than or equal to fifteen
+ ⓑ seven is equal to twelve minus five
+ ⓒ fifteen divided by three is less than eight
+ ⓓ #emph[y] minus three is greater than six
]
]

#examplebox("Example 3")[][
The information in  compares the fuel economy in miles-per-gallon (mpg) of several cars. Write the appropriate symbol #math.equation(block: false, alt: "= , < , or >")[$"=" , "<" , "or" #h(0.2em) ">"$] in each expression to compare the fuel economy of the cars.

#figure(figph[This table has two rows and six columns. The first column is a header column and it labels each row The first row is labeled “Car” and the second “Fuel economy (mpg)”. To the right of the ‘Car’ row are the labels: “Prius”, “Mini Cooper”, “Toyota Corolla”, “Versa”, “Honda Fit”. Each of these columns contains an image of the labeled car model. To the right of the “Fuel economy (mpg)” row are the algebraic equations: the letter p, the equals symbol, the number forty-eight; the letter m, the equals symbol, the number twenty-seven; the letter c, the equals symbol, the number twenty-eight; the letter v, the equals symbol, the number twenty-six; and the letter f, the equals symbol, the number twenty-seven.], alt: "This table has two rows and six columns. The first column is a header column and it labels each row The first row is labeled “Car” and the second “Fuel economy (mpg)”. To the right of the ‘Car’ row are the labels: “Prius”, “Mini Cooper”, “Toyota Corolla”, “Versa”, “Honda Fit”. Each of these columns contains an image of the labeled car model. To the right of the “Fuel economy (mpg)” row are the algebraic equations: the letter p, the equals symbol, the number forty-eight; the letter m, the equals symbol, the number twenty-seven; the letter c, the equals symbol, the number twenty-eight; the letter v, the equals symbol, the number twenty-six; and the letter f, the equals symbol, the number twenty-seven.", caption: [(credit: modification of work by Bernard Goldbach, Wikimedia Commons)])

+ ⓐ MPG of Prius\_\_\_\_\_ MPG of Mini Cooper
+ ⓑ MPG of Versa\_\_\_\_\_ MPG of Fit
+ ⓒ MPG of Mini Cooper\_\_\_\_\_ MPG of Fit
+ ⓓ MPG of Corolla\_\_\_\_\_ MPG of Versa
+ ⓔ MPG of Corolla\_\_\_\_\_ MPG of Prius

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓐ], []),
  [], [MPG of Prius\_\_\_\_MPG of Mini Cooper],
  [Find the values in the chart.], [48\_\_\_\_27],
  [Compare.], [48 \> 27],
  [], [MPG of Prius \> MPG of Mini Cooper],
))

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓑ], []),
  [], [MPG of Versa\_\_\_\_MPG of Fit],
  [Find the values in the chart.], [26\_\_\_\_27],
  [Compare.], [26 \< 27],
  [], [MPG of Versa \< MPG of Fit],
))

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓒ], []),
  [], [MPG of Mini Cooper\_\_\_\_MPG of Fit],
  [Find the values in the chart.], [27\_\_\_\_27],
  [Compare.], [27 = 27],
  [], [MPG of Mini Cooper = MPG of Fit],
))

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓓ], []),
  [], [MPG of Corolla\_\_\_\_MPG of Versa],
  [Find the values in the chart.], [28\_\_\_\_26],
  [Compare.], [28 \> 26],
  [], [MPG of Corolla \> MPG of Versa],
))

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓔ], []),
  [], [MPG of Corolla\_\_\_\_MPG of Prius],
  [Find the values in the chart.], [28\_\_\_\_48],
  [Compare.], [28 \< 48],
  [], [MPG of Corolla \< MPG of Prius],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Use  to fill in the appropriate #math.equation(block: false, alt: "symbol , = , < , or > .")[$"symbol" , "=" , "<" , "or" #h(0.2em) ">" .$]

+ ⓐ MPG of Prius\_\_\_\_\_MPG of Versa
+ ⓑ MPG of Mini Cooper\_\_\_\_\_ MPG of Corolla

#solutionbox[
+ ⓐ \>
+ ⓑ \<
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Use  to fill in the appropriate #math.equation(block: false, alt: "symbol , = , < , or > .")[$"symbol" , "=" , "<" , "or" #h(0.2em) ">" .$]

+ ⓐ MPG of Fit\_\_\_\_\_ MPG of Prius
+ ⓑ MPG of Corolla \_\_\_\_\_ MPG of Fit

#solutionbox[
+ ⓐ \<
+ ⓑ \>
]
]

Grouping symbols in algebra are much like the commas, colons, and other punctuation marks in written language. They indicate which expressions are to be kept together and separate from other expressions.  lists three of the most commonly used grouping symbols in algebra.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  [#strong[Common Grouping Symbols]], [],
  [parentheses], [#math.equation(block: false, alt: "open parenthesis close parenthesis")[$( #h(0.5em) )$]],
  [brackets], [#math.equation(block: false, alt: "[ ]")[$[ #h(0.5em) ]$]],
  [braces], [#math.equation(block: false, alt: "{ }")[$\{ #h(0.5em) \}$]],
))

Here are some examples of expressions that include 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 ] }")[$8 ( 14 − 8 ) #h(4em) 21 − 3 [ 2 + 4 ( 9 − 8 ) ] #h(4em) 24 ÷ \{ 13 − 2 [ 1 ( 6 − 5 ) + 4 ] \}$]
=== Identify Expressions and Equations

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]. An #strong[expression] is like a phrase. Here are some examples of expressions and how they relate to word phrases:

#figure(table(
  columns: 3, align: left, inset: 6pt,
  table.header([Expression], [Words], [Phrase]),
  [#math.equation(block: false, alt: "3 plus 5")[$3 + 5$]], [#math.equation(block: false, alt: "3 plus 5")[$3 #h(0.2em) "plus" #h(0.2em) 5$]], [the sum of three and five],
  [#math.equation(block: false, alt: "n minus 1")[$n − 1$]], [#math.equation(block: false, alt: "n")[$n$] minus one], [the difference of #math.equation(block: false, alt: "n")[$n$] and one],
  [#math.equation(block: false, alt: "6 times 7")[$6 · 7$]], [#math.equation(block: false, alt: "6 times 7")[$6 #h(0.2em) "times" #h(0.2em) 7$]], [the product of six and seven],
  [#math.equation(block: false, alt: "the fraction x over y")[$frac(x, y)$]], [#math.equation(block: false, alt: "x")[$x$] divided by #math.equation(block: false, alt: "y")[$y$]], [the quotient of #math.equation(block: false, alt: "x")[$x$] and #math.equation(block: false, alt: "y")[$y$]],
))

Notice that the phrases do not form a complete sentence because the phrase does not have a verb. An #strong[equation] is two expressions linked with an 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. Here are some examples of equations:

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Equation], [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")[$n − 1 = 14$]], [#math.equation(block: false, alt: "n")[$n$] 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")[$x = 53$]], [#math.equation(block: false, alt: "x")[$x$] is equal to fifty-three.],
  [#math.equation(block: false, alt: "y plus 9 equals 2 y minus 3")[$y + 9 = 2 y − 3$]], [#math.equation(block: false, alt: "y")[$y$] plus nine is equal to two #math.equation(block: false, alt: "y")[$y$] minus three.],
))

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Expressions and Equations]

An #strong[expression] is a number, a variable, or a combination of numbers and variables and operation symbols.

An #strong[equation] is made up of two expressions connected by an equal sign.
]

#examplebox("Example 4")[][
Determine if each is an expression or an equation:

+ ⓐ #math.equation(block: false, alt: "16 minus 6 equals 10")[$#h(0.2em) 16 − 6 = 10$]
+ ⓑ #math.equation(block: false, alt: "4 times 2 plus 1")[$#h(0.2em) 4 · 2 + 1$]
+ ⓒ #math.equation(block: false, alt: "x divided by 25")[$#h(0.2em) x ÷ 25$]
+ ⓓ #math.equation(block: false, alt: "y plus 8 equals 40")[$#h(0.2em) y + 8 = 40$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓐ #math.equation(block: false, alt: "16 minus 6 equals 10")[$#h(0.2em) 16 − 6 = 10$]], [This is an equation—two expressions are connected with an equal sign.]),
  [ⓑ #math.equation(block: false, alt: "4 times 2 plus 1")[$#h(0.2em) 4 · 2 + 1$]], [This is an expression—no equal sign.],
  [ⓒ #math.equation(block: false, alt: "x divided by 25")[$#h(0.2em) x ÷ 25$]], [This is an expression—no equal sign.],
  [ⓓ #math.equation(block: false, alt: "y plus 8 equals 40")[$#h(0.2em) y + 8 = 40$]], [This is an 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: "23 plus 6 equals 29")[$#h(0.2em) 23 + 6 = 29 #h(0.4em)$]
+ ⓑ #math.equation(block: false, alt: "7 times 3 minus 7")[$#h(0.2em) 7 · 3 − 7$]

#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 divided by 14")[$#h(0.2em) y ÷ 14 #h(0.4em)$]
+ #math.equation(block: false, alt: "x minus 6 equals 21")[$#h(0.2em) x − 6 = 21$]

#solutionbox[
+ ⓐ expression
+ ⓑ equation
]
]


=== Simplify Expressions with Exponents

To simplify a numerical 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 #math.equation(block: false, alt: "8")[$8$] and then add the #math.equation(block: false, alt: "1")[$1$] to get #math.equation(block: false, alt: "9 .")[$9 .$] 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")[$4 · 2 + 1$]#math.equation(block: true, alt: "8 plus 1")[$8 + 1$]#math.equation(block: true, alt: "9")[$9$]Suppose we have the expression #math.equation(block: false, alt: "2 times 2 times 2 times 2 times 2 times 2 times 2 times 2 times 2 .")[$2 · 2 · 2 · 2 · 2 · 2 · 2 · 2 · 2 .$] We could write this more compactly using exponential notation. Exponential notation is used in algebra to represent a quantity multiplied by itself several times. 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 #math.equation(block: false, alt: "2")[$2$] is called the #strong[base] and the #math.equation(block: false, alt: "3")[$3$] is called the exponent. The exponent tells us how many factors of the base we have to multiply.

#figure(figph[The image shows the number two with the number three, in superscript, to the right of the two. The number two is labeled as “base” and the number three is labeled as “exponent”.], alt: "The image shows the number two with the number three, in superscript, to the right of the two. The number two is labeled as “base” and the number three is labeled as “exponent”.", caption: none)

#math.equation(block: true, alt: "means multiply three factors of 2")[$"means multiply three factors of 2"$]We say #math.equation(block: false, alt: "2 cubed")[$2^(3)$] is in exponential notation and #math.equation(block: false, alt: "2 times 2 times 2")[$2 · 2 · 2$] is in expanded notation.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Exponential Notation]

For any expression #math.equation(block: false, alt: "a to the power n , a")[$a^(n) , a$] is a factor multiplied by itself #math.equation(block: false, alt: "n")[$n$] times if #math.equation(block: false, alt: "n")[$n$] is a positive integer.

#math.equation(block: true, alt: "a to the power n means multiply n factors of a")[$a^(n) "means multiply" #h(0.2em) n #h(0.2em) "factors of" #h(0.2em) a$]#figure(figph[At the top of the image is the letter a with the letter n, in superscript, to the right of the a. The letter a is labeled as “base” and the letter n is labeled as “exponent”. Below this is the letter a with the letter n, in superscript, to the right of the a set equal to n factors of a.], alt: "At the top of the image is the letter a with the letter n, in superscript, to the right of the a. The letter a is labeled as “base” and the letter n is labeled as “exponent”. Below this is the letter a with the letter n, in superscript, to the right of the a set equal to n factors of a.", caption: none)

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.
]

For powers of #math.equation(block: false, alt: "n equals 2")[$n = 2$] and #math.equation(block: false, alt: "n equals 3 ,")[$n = 3 ,$] we have special names.

#math.equation(block: true, alt: "a squared is read as \" a squared\"; a cubed is read as \" a cubed\"")[$a^(2) #h(0.2em) "is read as" #h(0.2em) "\"" a #h(0.2em) "squared\"" \ a^(3) #h(0.2em) "is read as" #h(0.2em) "\"" a #h(0.2em) "cubed\""$]lists some examples of expressions written in exponential notation.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Exponential Notation], [In Words]),
  [#math.equation(block: false, alt: "7 squared")[$7^(2)$]], [#math.equation(block: false, alt: "7")[$7$] to the second power, or #math.equation(block: false, alt: "7")[$7$] squared],
  [#math.equation(block: false, alt: "5 cubed")[$5^(3)$]], [#math.equation(block: false, alt: "5")[$5$] to the third power, or #math.equation(block: false, alt: "5")[$5$] cubed],
  [#math.equation(block: false, alt: "9 to the power 4")[$9^(4)$]], [#math.equation(block: false, alt: "9")[$9$] 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 5")[][
Write each expression in exponential form:

+ ⓐ #math.equation(block: false, alt: "16 times 16 times 16 times 16 times 16 times 16 times 16")[$#h(0.2em) 16 · 16 · 16 · 16 · 16 · 16 · 16$]
+ ⓑ #math.equation(block: false, alt: "9 times 9 times 9 times 9 times 9")[$#h(0.2em) 9 · 9 · 9 · 9 · 9$]
+ ⓒ #math.equation(block: false, alt: "x times x times x times x")[$#h(0.2em) x · x · x · x$]
+ ⓓ #math.equation(block: false, alt: "a times a times a times a times a times a times a times a")[$#h(0.2em) a · a · a · a · a · a · a · a$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓐ The base 16 is a factor 7 times.], [#math.equation(block: false, alt: "16 to the power 7")[$16^(7)$]]),
  [ⓑ The base 9 is a factor 5 times.], [#math.equation(block: false, alt: "9 to the power 5")[$9^(5)$]],
  [ⓒ The base #math.equation(block: false, alt: "x")[$x$] is a factor 4 times.], [#math.equation(block: false, alt: "x to the power 4")[$x^(4)$]],
  [ⓓ The base #math.equation(block: false, alt: "a")[$a$] is a factor 8 times.], [#math.equation(block: false, alt: "a to the power 8")[$a^(8)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Write each expression in exponential form:

#math.equation(block: true, alt: "41 times 41 times 41 times 41 times 41")[$41 · 41 · 41 · 41 · 41$]

#solutionbox[
41#super[5]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Write each expression in exponential form:

#math.equation(block: true, alt: "7 times 7 times 7 times 7 times 7 times 7 times 7 times 7 times 7")[$7 · 7 · 7 · 7 · 7 · 7 · 7 · 7 · 7$]

#solutionbox[
7#super[9]
]
]

#examplebox("Example 6")[][
Write each exponential expression in expanded form:

+ ⓐ #math.equation(block: false, alt: "8 to the power 6")[$#h(0.2em) 8^(6) #h(0.2em)$]
+ ⓑ #math.equation(block: false, alt: "x to the power 5")[$#h(0.2em) x^(5)$]

#solutionbox[
ⓐ The base is #math.equation(block: false, alt: "8")[$8$] and the exponent is #math.equation(block: false, alt: "6 ,")[$6 ,$] so #math.equation(block: false, alt: "8 to the power 6")[$8^(6)$] means #math.equation(block: false, alt: "8 times 8 times 8 times 8 times 8 times 8")[$8 · 8 · 8 · 8 · 8 · 8$]

ⓑ The base is #math.equation(block: false, alt: "x")[$x$] and the exponent is #math.equation(block: false, alt: "5 ,")[$5 ,$] so #math.equation(block: false, alt: "x to the power 5")[$x^(5)$] means #math.equation(block: false, alt: "x times x times x times x times x")[$x · x · x · x · x$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Write each exponential expression in expanded form:

+ ⓐ #math.equation(block: false, alt: "4 to the power 8")[$#h(0.2em) 4^(8)$]
+ ⓑ #math.equation(block: false, alt: "a to the power 7")[$#h(0.2em) a^(7)$]

#solutionbox[
+ ⓐ 4 · 4 · 4 · 4 · 4 · 4 · 4 · 4
+ ⓑ #emph[a] · #emph[a] · #emph[a] · #emph[a] · #emph[a] · #emph[a] · #emph[a]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Write each exponential expression in expanded form:

+ ⓐ #math.equation(block: false, alt: "8 to the power 8")[$attach(#h(0.2em) 8, t: 8) #h(0.4em)$]
+ ⓑ #math.equation(block: false, alt: "b to the power 6")[$#h(0.2em) b^(6)$]

#solutionbox[
+ ⓐ 8 · 8 · 8 · 8 · 8 · 8 · 8 · 8
+ ⓑ #emph[b] · #emph[b] · #emph[b] · #emph[b] · #emph[b] · #emph[b]
]
]

To simplify an exponential expression without using a calculator, we write it in expanded form and then multiply the factors.

#examplebox("Example 7")[][
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")[$9 ⋅ 3 ⋅ 3$]],
  [], [#math.equation(block: false, alt: "27 times 3")[$27 ⋅ 3$]],
  [Multiply.], [#math.equation(block: false, alt: "81")[$81$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify:

+ ⓐ #math.equation(block: false, alt: "5 cubed")[$#h(0.2em) 5^(3) #h(0.4em)$]
+ ⓑ #math.equation(block: false, alt: "1 to the power 7")[$#h(0.2em) 1^(7)$]

#solutionbox[
+ ⓐ 125
+ ⓑ 1
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify:

+ ⓐ #math.equation(block: false, alt: "7 squared")[$#h(0.2em) 7^(2) #h(0.4em)$]
+ ⓑ #math.equation(block: false, alt: "0 to the power 5")[$#h(0.2em) 0^(5)$]

#solutionbox[
+ ⓐ 49
+ ⓑ 0
]
]


=== Simplify Expressions Using the Order of Operations

We’ve introduced most of the symbols and notation used in algebra, but now we need to clarify the #strong[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")[$4 + 3 · 7$]#math.equation(block: true, alt: "Some students say it simplifies to 49., , Some students say it simplifies to 25.; 4 plus 3 times 7; Since 4 plus 3 gives 7., , 7 times 7; And 7 times 7 is 49., , 49, 4 plus 3 times 7; Since 3 times 7 is 21., 4 plus 21; And 21 plus 4 makes 25., 25")[$"Some students say it simplifies to 49." & #h(2em) &  & "Some students say it simplifies to 25." \  &  & 4 + 3 · 7 \ "Since" #h(0.2em) 4 + 3 #h(0.2em) "gives 7." & #h(2em) & 7 · 7 \ "And" #h(0.2em) 7 · 7 #h(0.2em) "is 49." & #h(2em) & 49 &  &  &  &  & 4 + 3 · 7 \ #h(0.2em) "Since" #h(0.2em) 3 · 7 #h(0.2em) "is 21." &  & 4 + 21 \ #h(0.2em) "And" #h(0.2em) 21 + 4 #h(0.2em) "makes 25." &  & 25$]Imagine the confusion that could result if every problem had several different correct answers. The same expression should give the same result. So mathematicians established some guidelines called the order of operations, which outlines the order in which parts of an expression must be simplified.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Order of Operations]

When simplifying mathematical expressions perform the operations in the following order:

1. #strong[P]arentheses and other Grouping Symbols

- Simplify all expressions inside the parentheses or other grouping symbols, working on the innermost parentheses first.

2. #strong[E]xponents

- Simplify all expressions with exponents.

3. #strong[M]ultiplication and #strong[D]ivision

- Perform all multiplication and division in order from left to right. These operations have equal priority.

4. #strong[A]ddition 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. #strong[P]lease #strong[E]xcuse #strong[M]y #strong[D]ear #strong[A]unt #strong[S]ally.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  [#strong[Order of Operations]], [],
  [#strong[P]lease], [#strong[P]arentheses],
  [#strong[E]xcuse], [#strong[E]xponents],
  [#strong[M]y #strong[D]ear], [#strong[M]ultiplication and #strong[D]ivision],
  [#strong[A]unt #strong[S]ally], [#strong[A]ddition and #strong[S]ubtraction],
))

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.

#examplebox("Example 8")[][
Simplify the expressions:

+ ⓐ #math.equation(block: false, alt: "4 plus 3 times 7")[$#h(0.2em) 4 + 3 · 7 #h(0.2em)$]
+ ⓑ #math.equation(block: false, alt: "open parenthesis 4 plus 3 close parenthesis times 7")[$#h(0.2em) ( 4 + 3 ) · 7$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓐ], []),
  [], [#figure(figph[A mathematical expression showing 4 plus 3 times 7, represented as 4 + 3  . 7.], alt: "A mathematical expression showing 4 plus 3 times 7, represented as 4 + 3 . 7.", caption: none)],
  [Are there any #emph[p]arentheses? No.], [],
  [Are there any #emph[e]xponents? No.], [],
  [Is there any #emph[m]ultiplication or #emph[d]ivision? Yes.], [],
  [Multiply first.], [#figure(figph[The image shows the mathematical expression '4 + 3 ⋅ 7'. The numbers '3' and '7' are in red, while '4' and the plus sign are in black.], alt: "The image shows the mathematical expression '4 + 3 ⋅ 7'. The numbers '3' and '7' are in red, while '4' and the plus sign are in black.", caption: none)],
  [Add.], [#figure(figph[The mathematical expression '4 + 21' is displayed against a white background.], alt: "The mathematical expression '4 + 21' is displayed against a white background.", caption: none)],
  [], [#figure(figph[The number 25 is prominently displayed on a plain white background.], alt: "The number 25 is prominently displayed on a plain white background.", caption: none)],
))

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓑ], []),
  [], [#figure(figph[The mathematical expression (4 + 3) multiplied by 7, clearly displaying the order of operations with parentheses around the addition.], alt: "The mathematical expression (4 + 3) multiplied by 7, clearly displaying the order of operations with parentheses around the addition.", caption: none)],
  [Are there any #emph[p]arentheses? Yes.], [#figure(figph[A mathematical expression reads (4 + 3) \* 7, with the numbers 4 and 3, and the plus sign in red, enclosed in red parentheses, followed by a black multiplication dot and the number 7.], alt: "A mathematical expression reads (4 + 3) * 7, with the numbers 4 and 3, and the plus sign in red, enclosed in red parentheses, followed by a black multiplication dot and the number 7.", caption: none)],
  [Simplify inside the parentheses.], [#figure(figph[The image shows a dark gray numerical expression on a white background, featuring the number seven enclosed in parentheses immediately followed by another number seven, appearing as (7)7.], alt: "The image shows a dark gray numerical expression on a white background, featuring the number seven enclosed in parentheses immediately followed by another number seven, appearing as (7)7.", caption: none)],
  [Are there any #emph[e]xponents? No.], [],
  [Is there any #emph[m]ultiplication or #emph[d]ivision? Yes.], [],
  [Multiply.], [#figure(figph[The number 49 is prominently displayed in the center of the image against a white background.], alt: "The number 49 is prominently displayed in the center of the image against a white background.", caption: none)],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify the expressions:

+ ⓐ #math.equation(block: false, alt: "12 minus 5 times 2")[$#h(0.2em) 12 − 5 · 2 #h(0.4em)$]
+ ⓑ #math.equation(block: false, alt: "open parenthesis 12 minus 5 close parenthesis times 2")[$#h(0.2em) ( 12 − 5 ) · 2$]

#solutionbox[
+ ⓐ 2
+ ⓑ  14
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify the expressions:

+ ⓐ #math.equation(block: false, alt: "8 plus 3 times 9")[$#h(0.2em) 8 + 3 · 9 #h(0.4em)$]
+ ⓑ #math.equation(block: false, alt: "open parenthesis 8 plus 3 close parenthesis times 9")[$( 8 + 3 ) · 9$]

#solutionbox[
+ ⓐ 35
+ ⓑ 99
]
]

#examplebox("Example 9")[][
Simplify:

+ ⓐ #math.equation(block: false, alt: "18 divided by 9 times 2")[$#h(0.2em) 18 ÷ 9 · 2 #h(0.2em)$]
+ ⓑ #math.equation(block: false, alt: "18 times 9 divided by 2")[$#h(0.2em) 18 · 9 ÷ 2$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓐ], []),
  [], [#figure(figph[A mathematical expression displays the numbers 18, 9, and 2, separated by a division symbol and a multiplication dot, reading as 18 ÷ 9 ⋅ 2.], alt: "A mathematical expression displays the numbers 18, 9, and 2, separated by a division symbol and a multiplication dot, reading as 18 ÷ 9 ⋅ 2.", caption: none)],
  [Are there any #emph[p]arentheses? No.], [],
  [Are there any #emph[e]xponents? No.], [],
  [Is there any #emph[m]ultiplication or #emph[d]ivision? Yes.], [],
  [Multiply and divide from left to right. Divide.], [#figure(figph[The image displays the number '2.2', with the first '2' in a red hue, followed by a black dot and another '2' in black, against a clean white background.], alt: "The image displays the number '2.2', with the first '2' in a red hue, followed by a black dot and another '2' in black, against a clean white background.", caption: none)],
  [Multiply.], [#figure(figph[The number four.], alt: "The number four.", caption: none)],
))

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓑ], []),
  [], [#figure(figph[The mathematical expression '18   9 ÷ 2' is shown, representing the multiplication of eighteen by nine, followed by the division of the product by two. The calculation yields a result of 81.], alt: "The mathematical expression '18 9 ÷ 2' is shown, representing the multiplication of eighteen by nine, followed by the division of the product by two. The calculation yields a result of 81.", caption: none)],
  [Are there any #emph[p]arentheses? No.], [],
  [Are there any #emph[e]xponents? No.], [],
  [Is there any #emph[m]ultiplication or #emph[d]ivision? Yes.], [],
  [Multiply and divide from left to right.], [],
  [Multiply.], [#figure(figph[A mathematical expression displays '162 ÷ 2', with the number 162 rendered in red and the division symbol and the number 2 in black against a white background.], alt: "A mathematical expression displays '162 ÷ 2', with the number 162 rendered in red and the division symbol and the number 2 in black against a white background.", caption: none)],
  [Divide.], [#figure(figph[The number 81 is shown in a bold, black font on a plain white background.], alt: "The number 81 is shown in a bold, black font on a plain white background.", caption: none)],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify:

#math.equation(block: true, alt: "42 divided by 7 times 3")[$42 ÷ 7 · 3$]

#solutionbox[
18
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify:

#math.equation(block: true, alt: "12 times 3 divided by 4")[$12 · 3 ÷ 4$]

#solutionbox[
9
]
]

#examplebox("Example 10")[][
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([], [#figure(figph[A mathematical expression featuring division, addition, and multiplication with parentheses: 18 ÷ 6 + 4(5 - 2).], alt: "A mathematical expression featuring division, addition, and multiplication with parentheses: 18 ÷ 6 + 4(5 - 2).", caption: none)]),
  [Parentheses? Yes, subtract first.], [#figure(figph[A mathematical expression reads '18 divided by 6 plus 4 times 3', or 18 / 6 + 4(3). The number '3' is highlighted in red, enclosed in parentheses.], alt: "A mathematical expression reads '18 divided by 6 plus 4 times 3', or 18 / 6 + 4(3). The number '3' is highlighted in red, enclosed in parentheses.", caption: none)],
  [Exponents? No.], [],
  [Multiplication or division? Yes.], [],
  [Divide first because we multiply and divide left to right.], [#figure(figph[A mathematical expression '3 + 4(3)' is displayed on a white background. The number 3 in '3 +' is colored red, while the rest of the expression is black.], alt: "A mathematical expression '3 + 4(3)' is displayed on a white background. The number 3 in '3 +' is colored red, while the rest of the expression is black.", caption: none)],
  [Any other multiplication or division? Yes.], [],
  [Multiply.], [#figure(figph[A mathematical expression '3 + 12' is displayed on a white background. The number '3' and the plus sign are black, while the number '12' is colored red.], alt: "A mathematical expression '3 + 12' is displayed on a white background. The number '3' and the plus sign are black, while the number '12' is colored red.", caption: none)],
  [Any other multiplication or division? No.], [],
  [Any addition or subtraction? Yes.], [#figure(figph[The number 15 is prominently displayed in the center of a plain white background.], alt: "The number 15 is prominently displayed in the center of a plain white background.", caption: none)],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify:

#math.equation(block: true, 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: true, 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 11")[][
#math.equation(block: true, alt: "Simplify: 5 plus 2 cubed plus 3 [ 6 minus 3 open parenthesis 4 minus 2 close parenthesis ] .")[$"Simplify:" #h(0.2em) 5 + 2^(3) + 3 [ 6 − 3 ( 4 − 2 ) ] .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A mathematical expression showing the order of operations: 5 + 2^3 + 3\[6 - 3(4 - 2)\].], alt: "A mathematical expression showing the order of operations: 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 expression reads 5 + 2^3 + 3\[6 - 3(4 - 2)\]. The subtraction within the inner parentheses, '4 - 2,' is highlighted in red.], alt: "A mathematical expression reads 5 + 2^3 + 3[6 - 3(4 - 2)]. The subtraction within the inner parentheses, '4 - 2,' is highlighted in red.", caption: none)],
  [Subtract.], [#figure(figph[A mathematical expression displaying 5 + 2^3 + 3\[6 - 3(2)\], with the '3(2)' part highlighted in red.], alt: "A mathematical expression displaying 5 + 2^3 + 3[6 - 3(2)], with the '3(2)' part highlighted in red.", caption: none)],
  [Continue inside the brackets and multiply.], [#figure(figph[A mathematical expression reads '5 + 2^3 + 3\[6 - 6\]'. The final '6' inside the bracket is highlighted in red.], alt: "A mathematical expression reads '5 + 2^3 + 3[6 - 6]'. The final '6' inside the bracket is highlighted in red.", caption: none)],
  [Continue inside the brackets and subtract.], [#figure(figph[A mathematical expression displaying 5 + 2 cubed + 3 multiplied by 0, with the 0 highlighted in red within brackets.], alt: "A mathematical expression displaying 5 + 2 cubed + 3 multiplied by 0, with the 0 highlighted in red within brackets.", caption: none)],
  [The expression inside the brackets requires no further simplification.], [],
  [Are there any exponents? Yes.], [],
  [Simplify exponents.], [#figure(figph[A mathematical expression: 5 + 2^3 + 3\[0\]. The term 2^3 is highlighted in red.], alt: "A mathematical expression: 5 + 2^3 + 3[0]. The term 2^3 is highlighted in red.", caption: none)],
  [Is there any multiplication or division? Yes.], [],
  [Multiply.], [#figure(figph[A mathematical expression reads '5 + 8 + 3\[0\]' with the '3\[0\]' part highlighted in red.], alt: "A mathematical expression reads '5 + 8 + 3[0]' with the '3[0]' part highlighted in red.", caption: none)],
  [Is there any addition or subtraction? Yes.], [],
  [Add.], [#figure(figph[A mathematical expression '5 + 8 + 0' is displayed on a white background. The number '5' and the first '+' sign are red. The number '8' and the second '+' sign are also red. The number '0' is black.], alt: "A mathematical expression '5 + 8 + 0' is displayed on a white background. The number '5' and the first '+' sign are red. The number '8' and the second '+' sign are also red. The number '0' is black.", caption: none)],
  [Add.], [#figure(figph[The image displays the mathematical expression '13 + 0' rendered in red characters against a plain white background, showing a simple addition problem.], alt: "The image displays the mathematical expression '13 + 0' rendered in red characters against a plain white background, showing a simple addition problem.", caption: none)],
  [], [#figure(figph[The number 13 is displayed in black text on a plain white background, centered in the frame.], alt: "The number 13 is displayed in black text on a plain white background, centered in the frame.", caption: none)],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify:

#math.equation(block: true, 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: true, alt: "7 squared minus 2 [ 4 open parenthesis 5 plus 1 close parenthesis ]")[$7^(2) − 2 [ 4 ( 5 + 1 ) ]$]

#solutionbox[
1
]
]

#examplebox("Example 12")[][
Simplify: #math.equation(block: false, alt: "2 cubed plus 3 to the power 4 divided by 3 minus 5 squared .")[$2^(3) + 3^(4) ÷ 3 − 5^(2) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A mathematical expression reads '2^3 + 3^4 ÷ 3 - 5^2'.], alt: "A mathematical expression reads '2^3 + 3^4 ÷ 3 - 5^2'.", caption: none)]),
  [If an expression has several exponents, they may be simplified in the same step.], [],
  [Simplify exponents.], [#figure(figph[A mathematical expression displaying powers, addition, division, and subtraction: 2^3 + 3^4 ÷ 3 - 5^2.], alt: "A mathematical expression displaying powers, addition, division, and subtraction: 2^3 + 3^4 ÷ 3 - 5^2.", caption: none)],
  [Divide.], [#figure(figph[A mathematical expression '8 + 81 : 3 - 25' is displayed. The numbers 81 and 3, along with the division symbol, are highlighted in red.], alt: "A mathematical expression '8 + 81 : 3 - 25' is displayed. The numbers 81 and 3, along with the division symbol, are highlighted in red.", caption: none)],
  [Add.], [#figure(figph[A mathematical expression '8 + 27 - 25' is displayed, with the numbers 8 and 27, and the plus sign in red, while the minus sign and number 25 are in black.], alt: "A mathematical expression '8 + 27 - 25' is displayed, with the numbers 8 and 27, and the plus sign in red, while the minus sign and number 25 are in black.", caption: none)],
  [Subtract.], [#figure(figph[The image displays the numbers 35 and 25 separated by a minus sign, forming the mathematical expression '35 - 25' in a reddish-brown font against a white background.], alt: "The image displays the numbers 35 and 25 separated by a minus sign, forming the mathematical expression '35 - 25' in a reddish-brown font against a white background.", caption: none)],
  [], [#figure(figph[The number 10 is prominently displayed in the upper right corner of a clean, white background, rendered in a bold, dark font.], alt: "The number 10 is prominently displayed in the upper right corner of a clean, white background, rendered in a bold, dark font.", caption: none)],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify:

#math.equation(block: true, alt: "3 squared plus 2 to the power 4 divided by 2 plus 4 cubed")[$3^(2) + 2^(4) ÷ 2 + 4^(3)$]

#solutionbox[
81
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify:

#math.equation(block: true, alt: "6 squared minus 5 cubed divided by 5 plus 8 squared")[$6^(2) − 5^(3) ÷ 5 + 8^(2)$]

#solutionbox[
75
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[ACCESS ADDITIONAL ONLINE RESOURCES]

- #link("https://openstaxcollege.org/l/24orderoperate")[Order of Operations]
- #link("https://openstaxcollege.org/l/24orderbasic")[Order of Operations – The Basics]
- #link("https://openstaxcollege.org/l/24Evalexpress")[Ex: Evaluate an Expression Using the Order of Operations]
- #link("https://openstaxcollege.org/l/24evalexpress3")[Example 3: Evaluate an Expression Using The Order of Operations]
]


=== Key Concepts

#figure(table(
  columns: 4, align: left, inset: 6pt,
  table.header([#strong[Operation]], [#strong[Notation]], [#strong[Say:]], [#strong[The result is…]]),
  [Addition], [#math.equation(block: false, alt: "a plus b")[$a + b$]], [#math.equation(block: false, alt: "a plus b")[$a #h(0.2em) "plus" #h(0.2em) b$]], [the sum of #math.equation(block: false, alt: "a")[$a$] and #math.equation(block: false, alt: "b")[$b$]],
  [Multiplication], [#math.equation(block: false, alt: "a times b , open parenthesis a close parenthesis open parenthesis b close parenthesis , open parenthesis a close parenthesis b , a open parenthesis b close parenthesis")[$a · b , ( a ) ( b ) , ( a ) b , a ( b )$]], [#math.equation(block: false, alt: "a times b")[$a #h(0.2em) "times" #h(0.2em) b$]], [The product of #math.equation(block: false, alt: "a")[$a$] and #math.equation(block: false, alt: "b")[$b$]],
  [Subtraction], [#math.equation(block: false, alt: "a minus b")[$a − b$]], [#math.equation(block: false, alt: "a minus b")[$a #h(0.2em) "minus" #h(0.2em) b$]], [the difference of #math.equation(block: false, alt: "a")[$a$] and #math.equation(block: false, alt: "b")[$b$]],
  [Division], [#math.equation(block: false, alt: "a divided by b , a / b , the fraction a over b , b a")[$a ÷ b , a / b , #h(0.2em) frac(a, b) , b ") " overline(a)$]], [#math.equation(block: false, alt: "a")[$a$] divided by #math.equation(block: false, alt: "b")[$b$]], [The quotient of #math.equation(block: false, alt: "a")[$a$] and #math.equation(block: false, alt: "b")[$b$]],
))

- #emph[Equality Symbol]
- #math.equation(block: false, alt: "a equals b")[$a = b$] is read as #math.equation(block: false, alt: "a")[$a$] is equal to #math.equation(block: false, alt: "b")[$b$]
- The symbol #math.equation(block: false, alt: "equals")[$=$] is called the equal sign.
- #emph[Inequality]
- #math.equation(block: false, alt: "a less than b")[$a < b$] is read #math.equation(block: false, alt: "a")[$a$] is less than #math.equation(block: false, alt: "b")[$b$]
- #math.equation(block: false, alt: "a")[$a$] is to the left of #math.equation(block: false, alt: "b")[$b$] on the number line
 #linebreak() #figure(figph[A number line with two points, 'a' and 'b', marked by vertical ticks. Point 'a' is to the left of point 'b', indicating that 'a' is a smaller value than 'b'.], alt: "A number line with two points, 'a' and 'b', marked by vertical ticks. Point 'a' is to the left of point 'b', indicating that 'a' is a smaller value than 'b'.", caption: none)
- #math.equation(block: false, alt: "a greater than b")[$a > b$] is read #math.equation(block: false, alt: "a")[$a$] is greater than #math.equation(block: false, alt: "b")[$b$]
- #math.equation(block: false, alt: "a")[$a$] is to the right of #math.equation(block: false, alt: "b")[$b$] on the number line
 #linebreak() #figure(figph[A number line shows points 'b' and 'a' marked by vertical ticks. Point 'b' is to the left of point 'a', indicating that b is less than a.], alt: "A number line shows points 'b' and 'a' marked by vertical ticks. Point 'b' is to the left of point 'a', indicating that b is less than a.", caption: none)

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Algebraic Notation], [Say]),
  [#math.equation(block: false, alt: "a equals b")[$a = b$]], [#math.equation(block: false, alt: "a")[$a$] is equal to #math.equation(block: false, alt: "b")[$b$]],
  [#math.equation(block: false, alt: "a not equal to b")[$a ≠ b$]], [#math.equation(block: false, alt: "a")[$a$] is not equal to #math.equation(block: false, alt: "b")[$b$]],
  [#math.equation(block: false, alt: "a less than b")[$a < b$]], [#math.equation(block: false, alt: "a")[$a$] is less than #math.equation(block: false, alt: "b")[$b$]],
  [#math.equation(block: false, alt: "a greater than b")[$a > b$]], [#math.equation(block: false, alt: "a")[$a$] is greater than #math.equation(block: false, alt: "b")[$b$]],
  [#math.equation(block: false, alt: "a less than or equal to b")[$a ≤ b$]], [#math.equation(block: false, alt: "a")[$a$] is less than or equal to #math.equation(block: false, alt: "b")[$b$]],
  [#math.equation(block: false, alt: "a greater than or equal to b")[$a ≥ b$]], [#math.equation(block: false, alt: "a")[$a$] is greater than or equal to #math.equation(block: false, alt: "b")[$b$]],
))

- #emph[Exponential Notation]
- For any expression #math.equation(block: false, alt: "a to the power n")[$a^(n)$] is a factor multiplied by itself #math.equation(block: false, alt: "n")[$n$] times, if #math.equation(block: false, alt: "n")[$n$] is a positive integer.
- #math.equation(block: false, alt: "a to the power n")[$a^(n)$] means multiply #math.equation(block: false, alt: "n")[$n$] factors of #math.equation(block: false, alt: "a")[$a$] #linebreak() #figure(figph[This image defines the components of an expression. In the example of a to the power of n, a is the base, while n is the exponent. a multiplied by a several times equals the n factors.], alt: "This image defines the components of an expression. In the example of a to the power of n, a is the base, while n is the exponent. a multiplied by a several times equals the n factors.", caption: none)
- The expression of #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 th")[$n "th"$] power.

#emph[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.


==== Practice Makes Perfect

#strong[Use Variables and Algebraic Symbols]

In the following exercises, translate from algebraic notation to words.

#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: "25 minus 7")[$25 − 7$]

#math.equation(block: true, alt: "5 times 6")[$5 · 6$]

#solutionbox[
5 times 6, the product of five and six
]

#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: "45 divided by 5")[$45 ÷ 5$]

#math.equation(block: true, alt: "x plus 8")[$x + 8$]

#solutionbox[
#emph[x] plus 8, the sum of #emph[x] and eight
]

#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: "42 greater than or equal to 27")[$42 ≥ 27$]

#math.equation(block: true, alt: "3 n equals 24")[$3 n = 24$]

#solutionbox[
3 times n equals 24, the product of three and n equals twenty-four
]

#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: "3 less than or equal to 20 divided by 4")[$3 ≤ 20 ÷ 4$]

#math.equation(block: true, alt: "a not equal to 7 times 4")[$a ≠ 7 · 4$]

#solutionbox[
#emph[a] is not equal to 7 times 4, #emph[a] is not equal to the product of seven and four
]

#math.equation(block: true, alt: "a not equal to 1 times 12")[$a ≠ 1 · 12$]

#strong[Identify Expressions and Equations]

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: "6 times 3 plus 5")[$6 · 3 + 5$]

#math.equation(block: true, alt: "x plus 7")[$x + 7$]

#solutionbox[
expression
]

#math.equation(block: true, alt: "x plus 9")[$x + 9$]

#math.equation(block: true, alt: "y minus 5 equals 25")[$y − 5 = 25$]

#solutionbox[
equation
]

#math.equation(block: true, alt: "y minus 8 equals 32")[$y − 8 = 32$]

#strong[Simplify Expressions with Exponents]

In the following exercises, write in exponential form.

#math.equation(block: true, alt: "3 times 3 times 3 times 3 times 3 times 3 times 3")[$3 · 3 · 3 · 3 · 3 · 3 · 3$]

#solutionbox[
3#super[7]
]

#math.equation(block: true, alt: "4 times 4 times 4 times 4 times 4 times 4")[$4 · 4 · 4 · 4 · 4 · 4$]

#math.equation(block: true, alt: "x times x times x times x times x")[$x · x · x · x · x$]

#solutionbox[
#emph[x]#super[5]
]

#math.equation(block: true, alt: "y times y times y times y times y times y")[$y · y · y · y · y · y$]

In the following exercises, write in expanded form.

#math.equation(block: true, alt: "5 cubed")[$5^(3)$]

#solutionbox[
#math.equation(block: true, alt: "5 times 5 times 5")[$5 · 5 · 5$]
]

#math.equation(block: true, alt: "8 cubed")[$8^(3)$]

#math.equation(block: true, alt: "2 to the power 8")[$2^(8)$]

#solutionbox[
#math.equation(block: true, alt: "2 times 2 times 2 times 2 times 2 times 2 times 2 times 2")[$2 · 2 · 2 · 2 · 2 · 2 · 2 · 2$]
]

#math.equation(block: true, alt: "10 to the power 5")[$10^(5)$]

#strong[Simplify Expressions Using the Order of Operations]

In the following exercises, simplify.

+ ⓐ #math.equation(block: false, alt: "3 plus 8 times 5")[$#h(0.2em) 3 + 8 · 5 #h(0.2em) #h(0.4em)$]
+ ⓑ #math.equation(block: false, alt: "(3+8) times 5")[$#h(0.2em) "(3+8)" · 5$]

#solutionbox[
+ ⓐ 43
+ ⓑ 55
]

+ ⓐ #math.equation(block: false, alt: "2 plus 6 times 3")[$#h(0.2em) 2 + 6 · 3 #h(0.2em) #h(0.4em)$]
+ ⓑ #math.equation(block: false, alt: "(2+6) times 3")[$#h(0.2em) "(2+6)" · 3$]

#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: "6 plus 10 / 2 plus 2")[$6 + 10 / 2 + 2$]

#solutionbox[
13
]

#math.equation(block: true, alt: "9 plus 12 / 3 plus 4")[$9 + 12 / 3 + 4$]

#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: "20 divided by open parenthesis 4 plus 6 close parenthesis times 5")[$20 ÷ ( 4 + 6 ) · 5$]

#solutionbox[
10
]

#math.equation(block: true, alt: "33 divided by open parenthesis 3 plus 8 close parenthesis times 2")[$33 ÷ ( 3 + 8 ) · 2$]

#math.equation(block: true, alt: "4 squared plus 5 squared")[$4^(2) + 5^(2)$]

#solutionbox[
41
]

#math.equation(block: true, alt: "3 squared plus 7 squared")[$3^(2) + 7^(2)$]

#math.equation(block: true, alt: "open parenthesis 4 plus 5 close parenthesis squared")[$attach(( 4 + 5 ), t: 2)$]

#solutionbox[
81
]

#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 ]")[$5 [ 2 + 4 ( 3 − 2 ) ]$]


==== Everyday Math

#strong[Basketball] In the 2014 NBA playoffs, the San Antonio Spurs beat the Miami Heat. The table below shows the heights of the starters on each team. Use this table to fill in the appropriate symbol #math.equation(block: false, alt: "(= , < , >) .")[$"(=" , "<" , ">)" .$]

#figure(table(
  columns: 5, align: left, inset: 6pt,
  [#strong[Spurs]], [#strong[Height]], [], [#strong[Heat]], [#strong[Height]],
  [Tim Duncan], [#math.equation(block: false, alt: "83″")[$"83″"$]], [], [Rashard Lewis], [#math.equation(block: false, alt: "82″")[$"82″"$]],
  [Boris Diaw], [#math.equation(block: false, alt: "80″")[$"80″"$]], [LeBron James], [#math.equation(block: false, alt: "80″")[$"80″"$]], [],
  [Kawhi Leonard], [#math.equation(block: false, alt: "79″")[$"79″"$]], [Chris Bosh], [#math.equation(block: false, alt: "83″")[$"83″"$]], [],
  [Tony Parker], [#math.equation(block: false, alt: "74″")[$"74″"$]], [Dwyane Wade], [#math.equation(block: false, alt: "76″")[$"76″"$]], [],
  [Danny Green], [#math.equation(block: false, alt: "78″")[$"78″"$]], [Ray Allen], [#math.equation(block: false, alt: "77″")[$"77″"$]], [],
))

+ ⓐ Height of Tim Duncan\_\_\_\_Height of Rashard Lewis
+ ⓑ Height of Boris Diaw\_\_\_\_Height of LeBron James
+ ⓒ Height of Kawhi Leonard\_\_\_\_Height of Chris Bosh
+ ⓓ Height of Tony Parker\_\_\_\_Height of Dwyane Wade
+ ⓔ Height of Danny Green\_\_\_\_Height of Ray Allen

#strong[Elevation] In Colorado there are more than #math.equation(block: false, alt: "50")[$50$] mountains with an elevation of over #math.equation(block: false, alt: "14,000 feet.")[$14,000 #h(0.2em) "feet."$] The table shows the ten tallest. Use this table to fill in the appropriate inequality symbol.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Mountain], [Elevation]),
  [Mt. Elbert], [#math.equation(block: false, alt: "14,433′")[$"14,433′"$]],
  [Mt. Massive], [#math.equation(block: false, alt: "14,421′")[$"14,421′"$]],
  [Mt. Harvard], [#math.equation(block: false, alt: "14,420′")[$"14,420′"$]],
  [Blanca Peak], [#math.equation(block: false, alt: "14,345′")[$"14,345′"$]],
  [La Plata Peak], [#math.equation(block: false, alt: "14,336′")[$"14,336′"$]],
  [Uncompahgre Peak], [#math.equation(block: false, alt: "14,309′")[$"14,309′"$]],
  [Crestone Peak], [#math.equation(block: false, alt: "14,294′")[$"14,294′"$]],
  [Mt. Lincoln], [#math.equation(block: false, alt: "14,286′")[$"14,286′"$]],
  [Grays Peak], [#math.equation(block: false, alt: "14,270′")[$"14,270′"$]],
  [Mt. Antero], [#math.equation(block: false, alt: "14,269′")[$"14,269′"$]],
))

+ ⓐ Elevation of La Plata Peak\_\_\_\_Elevation of Mt. Antero
+ ⓑ Elevation of Blanca Peak\_\_\_\_Elevation of Mt. Elbert
+ ⓒ Elevation of Gray’s Peak\_\_\_\_Elevation of Mt. Lincoln
+ ⓓ Elevation of Mt. Massive\_\_\_\_Elevation of Crestone Peak
+ ⓔ Elevation of Mt. Harvard\_\_\_\_Elevation of Uncompahgre Peak


==== Writing Exercises

Explain the difference between an expression and an equation.

Why is it important to use the order of operations to simplify an expression?


==== Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

#figure(figph[A self-assessment checklist for students to evaluate their understanding of algebraic concepts, including using variables, identifying expressions and equations, and simplifying expressions with exponents and order of operations.], alt: "A self-assessment checklist for students to evaluate their understanding of algebraic concepts, including using variables, identifying expressions and equations, and simplifying expressions with exponents and order of operations.", caption: none)

ⓑ If most of your checks were:

…confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.

…with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math, every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Whom can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?

…no—I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.