#set document(title: "8.6 Solve Rational Equations", 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")) == 8.6#h(0.6em)Solve Rational Equations #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Before you get started, take this readiness quiz. If you miss a problem, go back to the section listed and review the material. Solve: #math.equation(block: false, alt: "the fraction 1 over 6 x plus the fraction 1 over 2 equals the fraction 1 over 3 .")[$frac(1, 6) x + frac(1, 2) = frac(1, 3) .$] #linebreak() If you missed this problem, review . #solutionbox[ #math.equation(block: true, alt: "x equals minus 1")[$x = − 1$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve: #math.equation(block: false, alt: "n squared minus 5 n minus 36 equals 0 .")[$n^(2) − 5 n − 36 = 0 .$] #linebreak() If you missed this problem, review . #solutionbox[ #math.equation(block: true, alt: "n equals 9 , n equals minus 4")[$n = 9 , #h(0.2em) n = − 4$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve for #math.equation(block: false, alt: "y")[$y$] in terms of #math.equation(block: false, alt: "x")[$x$]: #math.equation(block: false, alt: "5 x plus 2 y equals 10")[$5 x + 2 y = 10$] for #math.equation(block: false, alt: "y .")[$y .$] #linebreak() If you missed this problem, review . #solutionbox[ #math.equation(block: true, alt: "y equals the fraction 10 minus 5 x over 2")[$y = frac(10 − 5 x, 2)$] ] ] After defining the terms #emph[expression] and #emph[equation] early in Foundations, we have used them throughout this book. We have #emph[simplified] many kinds of #emph[expressions] and #emph[solved] many kinds of #emph[equations]. We have simplified many rational expressions so far in this chapter. Now we will solve rational equations. The definition of a rational equation is similar to the definition of equation we used in Foundations. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Rational Equation] A #strong[rational equation] is two rational expressions connected by an equal sign. ] You must make sure to know the difference between rational expressions and rational equations. The equation contains an equal sign. #math.equation(block: true, alt: "Rational Expression, Rational Equation; the fraction 1 over 8 x plus the fraction 1 over 2, the fraction 1 over 8 x plus the fraction 1 over 2 equals the fraction 1 over 4; the fraction y plus 6 over y squared minus 36, the fraction y plus 6 over y squared minus 36 equals y plus 1; the fraction 1 over n minus 3 plus the fraction 1 over n plus 4, the fraction 1 over n minus 3 plus the fraction 1 over n plus 4 equals the fraction 15 over n squared plus n minus 12")[$"Rational Expression" & & & #h(5em) "Rational Equation" \ #h(1.95em) frac(1, 8) x + frac(1, 2) & & & #h(3.1em) frac(1, 8) x + frac(1, 2) = frac(1, 4) \ #h(2.5em) frac(y + 6, y^(2) − 36) & & & #h(4.65em) frac(y + 6, y^(2) − 36) = y + 1 \ #h(2.5em) frac(1, n − 3) + frac(1, n + 4) & & & #h(5em) frac(1, n − 3) + frac(1, n + 4) = frac(15, n^(2) + n − 12)$] === Solve Rational Equations We have already solved linear equations that contained fractions. We found the LCD of all the fractions in the equation and then multiplied both sides of the equation by the LCD to “clear” the fractions. Here is an example we did when we worked with linear equations: #figure(table( columns: 3, align: left, inset: 6pt, table.header([], [#figure(figph[A mathematical equation showing one-eighth x plus one-half equals one-fourth, or (1/8)x + (1/2) = (1/4).], alt: "A mathematical equation showing one-eighth x plus one-half equals one-fourth, or (1/8)x + (1/2) = (1/4).", caption: none)], [#figure(figph[The image shows 'LCD = 8' in black text on a white background, representing a mathematical or computational expression where the least common denominator (LCD) is equal to eight.], alt: "The image shows 'LCD = 8' in black text on a white background, representing a mathematical or computational expression where the least common denominator (LCD) is equal to eight.", caption: none)]), [We multiplied both sides by the LCD.], [#figure(figph[A mathematical equation shows '8(1/8x + 1/2) = 8(1/4)', with the number 8 and the parentheses highlighted in red.], alt: "A mathematical equation shows '8(1/8x + 1/2) = 8(1/4)', with the number 8 and the parentheses highlighted in red.", caption: none)], [], [Then we distributed.], [#figure(figph[A mathematical equation is displayed, showing '8 multiplied by 1/8x plus 8 multiplied by 1/2 equals 8 multiplied by 1/4' against a white background.], alt: "A mathematical equation is displayed, showing '8 multiplied by 1/8x plus 8 multiplied by 1/2 equals 8 multiplied by 1/4' against a white background.", caption: none)], [], [We simplified—and then we had an equation with no fractions.], [#figure(figph[A simple algebraic equation 'x + 4 = 2' is displayed in black text against a white background.], alt: "A simple algebraic equation 'x + 4 = 2' is displayed in black text against a white background.", caption: none)], [], [Finally, we solved that equation.], [#figure(figph[A mathematical equation is displayed on a white background: x + 4 - 4 = 2 - 4. The -4 on both sides of the equation is highlighted in red, indicating subtraction from both sides to solve for x.], alt: "A mathematical equation is displayed on a white background: x + 4 - 4 = 2 - 4. The -4 on both sides of the equation is highlighted in red, indicating subtraction from both sides to solve for x.", caption: none)], [], [], [#figure(figph[The image displays the equation 'X = -2' written in a bold, sans-serif font against a plain white background.], alt: "The image displays the equation 'X = -2' written in a bold, sans-serif font against a plain white background.", caption: none)], [], )) We will use the same strategy to solve rational equations. We will multiply both sides of the equation by the LCD. Then we will have an equation that does not contain rational expressions and thus is much easier for us to solve. But because the original equation may have a variable in a denominator we must be careful that we don’t end up with a solution that would make a denominator equal to zero. So before we begin solving a rational equation, we examine it first to find the values that would make any denominators zero. That way, when we solve a #strong[rational equation] we will know if there are any algebraic solutions we must discard. An algebraic solution to a rational equation that would cause any of the rational expressions to be undefined is called an #emph[extraneous solution]. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Extraneous Solution to a Rational Equation] An #strong[extraneous solution to a rational equation] is an algebraic solution that would cause any of the expressions in the original equation to be undefined. ] We note any possible extraneous solutions, #emph[c], by writing #math.equation(block: false, alt: "x not equal to c")[$x ≠ c$] next to the equation. #examplebox("Example 1")[How to Solve Equations with Rational Expressions][ Solve: #math.equation(block: false, alt: "the fraction 1 over x plus the fraction 1 over 3 equals the fraction 5 over 6 .")[$frac(1, x) + frac(1, 3) = frac(5, 6) .$] #solutionbox[ #figure(figph[The above image has 3 columns. It shows the steps to find an extraneous solution to a rational equation for the example 1 divided by x plus one-third equals five-sixths. Step one is to note any value of the variable that would make any denominator zero. If x equals 0, then I divided by x is undefined. So we’ll write x divided zero next to the equation to get 1 divided by x plus one-third equals five-sixths times x divided by zero.], alt: "The above image has 3 columns. It shows the steps to find an extraneous solution to a rational equation for the example 1 divided by x plus one-third equals five-sixths. Step one is to note any value of the variable that would make any denominator zero. If x equals 0, then I divided by x is undefined. So we’ll write x divided zero next to the equation to get 1 divided by x plus one-third equals five-sixths times x divided by zero.", caption: none) #figure(figph[Step two is to find the least common denominator of all denominators in the equation. Find the LCD of 1 divided by x one-third, and five-sixths. The x is 6 x.], alt: "Step two is to find the least common denominator of all denominators in the equation. Find the LCD of 1 divided by x one-third, and five-sixths. The x is 6 x.", caption: none) #figure(figph[Step three is to clear the fractions by multiplying both sides of the equation by the LCD. Multiply both sides of the equation by the LCD, 6 x to get 6 times 1 divided by x plus one-third equals 6 x times five-sixths. Use the Distributive Property to get 6 x times 1 divided by x plus 6 x times one-third equals 6 x times five-sixths. Simplify – and notice, no more fractions and we have 6 plus 2 x equals 5 x.], alt: "Step three is to clear the fractions by multiplying both sides of the equation by the LCD. Multiply both sides of the equation by the LCD, 6 x to get 6 times 1 divided by x plus one-third equals 6 x times five-sixths. Use the Distributive Property to get 6 x times 1 divided by x plus 6 x times one-third equals 6 x times five-sixths. Simplify – and notice, no more fractions and we have 6 plus 2 x equals 5 x.", caption: none) #figure(figph[Step 4 is to solve the resulting equation. Simplify to get 6 equals 3 x and 2 equals x.], alt: "Step 4 is to solve the resulting equation. Simplify to get 6 equals 3 x and 2 equals x.", caption: none) #figure(figph[Step 5 is to check. If any values found in Step 1 are algebraic solutions, discard them. Check any remaining solutions in the original equation. We did not get 0 as an algebraic solution. We substitute x equals 2 into the original equation to get one-half plus one-third equals five-sixths, then three-sixths plus two-sixths equals five-sixths and finally, five-sixths equal five-sixths.], alt: "Step 5 is to check. If any values found in Step 1 are algebraic solutions, discard them. Check any remaining solutions in the original equation. We did not get 0 as an algebraic solution. We substitute x equals 2 into the original equation to get one-half plus one-third equals five-sixths, then three-sixths plus two-sixths equals five-sixths and finally, five-sixths equal five-sixths.", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve: #math.equation(block: false, alt: "the fraction 1 over y plus the fraction 2 over 3 equals the fraction 1 over 5 .")[$frac(1, y) + frac(2, 3) = frac(1, 5) .$] #solutionbox[ #math.equation(block: true, alt: "minus the fraction 15 over 7")[$− frac(15, 7)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve: #math.equation(block: false, alt: "the fraction 2 over 3 plus the fraction 1 over 5 equals the fraction 1 over x .")[$frac(2, 3) + frac(1, 5) = frac(1, x) .$] #solutionbox[ #math.equation(block: true, alt: "the fraction 15 over 13")[$frac(15, 13)$] ] ] The steps of this method are shown below. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Solve equations with rational expressions.] + Note any value of the variable that would make any denominator zero. + Find the least common denominator of #emph[all] denominators in the equation. + Clear the fractions by multiplying both sides of the equation by the LCD. + Solve the resulting equation. + Check. - If any values found in Step 1 are algebraic solutions, discard them. - Check any remaining solutions in the original equation. ] We always start by noting the values that would cause any denominators to be zero. #examplebox("Example 2")[][ Solve: #math.equation(block: false, alt: "1 minus the fraction 5 over y equals minus the fraction 6 over y squared .")[$1 − frac(5, y) = − frac(6, y^(2)) .$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[A mathematical equation is shown with the expression 1 - 5/y = 6/y^2, displayed in a white background.], alt: "A mathematical equation is shown with the expression 1 - 5/y = 6/y^2, displayed in a white background.", caption: none)]), [Note any value of the variable that would make any denominator zero.], [#figure(figph[A mathematical equation is displayed: 1 minus 5 over y equals negative 6 over y squared, with the condition that y is not equal to 0.], alt: "A mathematical equation is displayed: 1 minus 5 over y equals negative 6 over y squared, with the condition that y is not equal to 0.", caption: none)], [Find the least common denominator of all denominators in the equation. The LCD is#math.equation(block: false, alt: "y squared")[$y^(2)$].], [], [Clear the fractions by multiplying both sides of the equation by the LCD.], [#figure(figph[An algebraic equation showing y squared times (1 minus 5 over y) equals y squared times (negative 6 over y squared).], alt: "An algebraic equation showing y squared times (1 minus 5 over y) equals y squared times (negative 6 over y squared).", caption: none)], [Distribute.], [#figure(figph[An algebraic equation is shown where y squared multiplied by 1 minus y squared multiplied by 5 over y equals y squared multiplied by negative 6 over y squared, with y squared in red.], alt: "An algebraic equation is shown where y squared multiplied by 1 minus y squared multiplied by 5 over y equals y squared multiplied by negative 6 over y squared, with y squared in red.", caption: none)], [Multiply.], [#figure(figph[A quadratic equation is displayed: y^2 - 5y = -6.], alt: "A quadratic equation is displayed: y^2 - 5y = -6.", caption: none)], [Solve the resulting equation. First write the quadratic equation in standard form.], [#figure(figph[The image displays the quadratic equation y^2 - 5y + 6 = 0, presented in a clear, standard mathematical format on a white background.], alt: "The image displays the quadratic equation y^2 - 5y + 6 = 0, presented in a clear, standard mathematical format on a white background.", caption: none)], [Factor.], [#figure(figph[A mathematical equation shown as (y-2)(y-3)=0, which is a quadratic equation in factored form.], alt: "A mathematical equation shown as (y-2)(y-3)=0, which is a quadratic equation in factored form.", caption: none)], [Use the Zero Product Property.], [#figure(figph[The image shows two mathematical equations separated by the word 'or': y-2=0 and y-3=0.], alt: "The image shows two mathematical equations separated by the word 'or': y-2=0 and y-3=0.", caption: none)], [Solve.], [#figure(figph[A mathematical expression states 'y = 2 or y = 3' in a black serif font on a plain white background.], alt: "A mathematical expression states 'y = 2 or y = 3' in a black serif font on a plain white background.", caption: none)], [Check.], [], [We did not get 0 as an algebraic solution.], [], [#figure(figph[Verification of solutions for an algebraic equation, showing y=2 and y=3 both satisfy 1 - 5/y = -6/y^2 through step-by-step substitution.], alt: "Verification of solutions for an algebraic equation, showing y=2 and y=3 both satisfy 1 - 5/y = -6/y^2 through step-by-step substitution.", caption: none)], [], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve: #math.equation(block: false, alt: "1 minus the fraction 2 over a equals the fraction 15 over a squared .")[$1 − frac(2, a) = frac(15, a^(2)) .$] #solutionbox[ #math.equation(block: true, alt: "5 , −3")[$5 , −3$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve: #math.equation(block: false, alt: "1 minus the fraction 4 over b equals the fraction 12 over b squared .")[$1 − frac(4, b) = frac(12, b^(2)) .$] #solutionbox[ #math.equation(block: true, alt: "6 , −2")[$6 , −2$] ] ] #examplebox("Example 3")[][ Solve: #math.equation(block: false, alt: "the fraction 5 over 3 u minus 2 equals the fraction 3 over 2 u .")[$frac(5, 3 u − 2) = frac(3, 2 u) .$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[A mathematical equation is displayed, showing the fraction 5 over the expression 3u minus 2, set equal to the fraction 3 over 2u.], alt: "A mathematical equation is displayed, showing the fraction 5 over the expression 3u minus 2, set equal to the fraction 3 over 2u.", caption: none)]), [Note any value of the variable that would make any denominator zero.], [#figure(figph[A mathematical equation shows 5 divided by (3u - 2) equals 3 divided by (2u), with the conditions that u is not equal to 2/3 and u is not equal to 0.], alt: "A mathematical equation shows 5 divided by (3u - 2) equals 3 divided by (2u), with the conditions that u is not equal to 2/3 and u is not equal to 0.", caption: none)], [Find the least common denominator of all denominators in the equation. The LCD is#math.equation(block: false, alt: "2 u open parenthesis 3 u minus 2 close parenthesis")[$2 u ( 3 u − 2 )$].], [], [Clear the fractions by multiplying both sides of the equation by the LCD.], [#figure(figph[An algebraic equation showing 2u(3u-2) multiplied by a fraction on both sides of the equality, with the fractions being 5/(3u-2) on the left and 3/(2u) on the right.], alt: "An algebraic equation showing 2u(3u-2) multiplied by a fraction on both sides of the equality, with the fractions being 5/(3u-2) on the left and 3/(2u) on the right.", caption: none)], [Remove common factors.], [#figure(figph[An algebraic equation illustrating the cancellation of common factors on both sides, a method used to simplify expressions and solve for the variable 'u'.], alt: "An algebraic equation illustrating the cancellation of common factors on both sides, a method used to simplify expressions and solve for the variable 'u'.", caption: none)], [Simplify.], [#figure(figph[A mathematical equation is displayed, showing '2u(5) = (3u-2)(3)'. The equation involves the variable 'u' and numerical constants, indicating a problem to be solved for 'u'.], alt: "A mathematical equation is displayed, showing '2u(5) = (3u-2)(3)'. The equation involves the variable 'u' and numerical constants, indicating a problem to be solved for 'u'.", caption: none)], [Multiply.], [#figure(figph[A mathematical equation is displayed on a white background: 10u = 9u - 6. The text is rendered in a black sans-serif font.], alt: "A mathematical equation is displayed on a white background: 10u = 9u - 6. The text is rendered in a black sans-serif font.", caption: none)], [Solve the resulting equation.], [#figure(figph[A white background features a mathematical expression, likely handwritten, that reads 'U = 6' in the center. The equation is rendered in a simple, dark, slightly faded script.], alt: "A white background features a mathematical expression, likely handwritten, that reads 'U = 6' in the center. The equation is rendered in a simple, dark, slightly faded script.", caption: none)], [We did not get 0 or #math.equation(block: false, alt: "the fraction 2 over 3")[$frac(2, 3)$] as algebraic solutions.], [], [#figure(figph[Checking the value u = -6 in the equation 5/(3u-2) = 3/(2u). The process shows substitution and simplification, resulting in -1/4 = -1/4, confirming that u = -6 is the correct solution for the equation.], alt: "Checking the value u = -6 in the equation 5/(3u-2) = 3/(2u). The process shows substitution and simplification, resulting in -1/4 = -1/4, confirming that u = -6 is the correct solution for the equation.", caption: none)], [], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve: #math.equation(block: false, alt: "the fraction 1 over x minus 1 equals the fraction 2 over 3 x .")[$frac(1, x − 1) = frac(2, 3 x) .$] #solutionbox[ #math.equation(block: true, alt: "−2")[$−2$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve: #math.equation(block: false, alt: "the fraction 3 over 5 n plus 1 equals the fraction 2 over 3 n .")[$frac(3, 5 n + 1) = frac(2, 3 n) .$] #solutionbox[ #math.equation(block: true, alt: "−2")[$−2$] ] ] When one of the denominators is a quadratic, remember to factor it first to find the LCD. #examplebox("Example 4")[][ Solve: #math.equation(block: false, alt: "the fraction 2 over p plus 2 plus the fraction 4 over p minus 2 equals the fraction p minus 1 over p squared minus 4 .")[$frac(2, p + 2) + frac(4, p − 2) = frac(p − 1, p^(2) − 4) .$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[A mathematical equation is displayed, showing a sum of two fractions on the left side equal to a fraction on the right side. The equation is (2/(p+2)) + (4/(p-2)) = (p-1)/(p^2-4).], alt: "A mathematical equation is displayed, showing a sum of two fractions on the left side equal to a fraction on the right side. The equation is (2/(p+2)) + (4/(p-2)) = (p-1)/(p^2-4).", caption: none)]), [Note any value of the variable that would make any denominator zero.], [#figure(figph[An algebraic equation is shown, featuring the sum of two fractions, 2/(p+2) and 4/(p-2), equaling a third fraction, (p-1)/((p+2)(p+2)), with the conditions p not equal to -2, p not equal to 2.], alt: "An algebraic equation is shown, featuring the sum of two fractions, 2/(p+2) and 4/(p-2), equaling a third fraction, (p-1)/((p+2)(p+2)), with the conditions p not equal to -2, p not equal to 2.", caption: none)], [Find the least common denominator of all denominators in the equation. The LCD is#math.equation(block: false, alt: "open parenthesis p plus 2 close parenthesis open parenthesis p minus 2 close parenthesis")[$( p + 2 ) ( p − 2 )$].], [], [Clear the fractions by multiplying both sides of the equation by the LCD.], [#figure(figph[An algebraic equation showing (p+2)(p-2) multiplied by the sum of two fractions on the left side, equaling (p+2)(p-2) multiplied by a single fraction on the right side. The common multiplier (p+2)(p-2) is in red.], alt: "An algebraic equation showing (p+2)(p-2) multiplied by the sum of two fractions on the left side, equaling (p+2)(p-2) multiplied by a single fraction on the right side. The common multiplier (p+2)(p-2) is in red.", caption: none)], [Distribute.], [#figure(figph[A mathematical equation where a common factor, (p+2)(p-2), highlighted in red, is multiplied across three terms involving fractions. This step is typically used to clear denominators in rational equations.], alt: "A mathematical equation where a common factor, (p+2)(p-2), highlighted in red, is multiplied across three terms involving fractions. This step is typically used to clear denominators in rational equations.", caption: none)], [Remove common factors.], [#figure(figph[An algebra problem demonstrating the solution of a rational equation by multiplying all terms by the least common denominator (p+2)(p-2). Red cross-outs show the cancellation of common factors.], alt: "An algebra problem demonstrating the solution of a rational equation by multiplying all terms by the least common denominator (p+2)(p-2). Red cross-outs show the cancellation of common factors.", caption: none)], [Simplify.], [#figure(figph[A mathematical equation is displayed, showing 2 multiplied by (p minus 2), plus 4 multiplied by (p plus 2), equals p minus 1.], alt: "A mathematical equation is displayed, showing 2 multiplied by (p minus 2), plus 4 multiplied by (p plus 2), equals p minus 1.", caption: none)], [Distribute.], [#figure(figph[A mathematical equation is displayed, reading '2p - 4 + 4p + 8 = p - 1' against a white background.], alt: "A mathematical equation is displayed, reading '2p - 4 + 4p + 8 = p - 1' against a white background.", caption: none)], [Solve.], [#figure(figph[A clear, well-lit image displays the algebraic equation 6p+4=p-1 centered against a plain white background, showing a simple linear equation that can be solved for the variable 'p'.], alt: "A clear, well-lit image displays the algebraic equation 6p+4=p-1 centered against a plain white background, showing a simple linear equation that can be solved for the variable 'p'.", caption: none)], [], [#figure(figph[An algebraic equation '5p = 5' is displayed in black text on a clean white background, indicating a simple mathematical problem to be solved.], alt: "An algebraic equation '5p = 5' is displayed in black text on a clean white background, indicating a simple mathematical problem to be solved.", caption: none)], [], [#figure(figph[The mathematical equation p=-1.], alt: "The mathematical equation p=-1.", caption: none)], [We did not get #math.equation(block: false, alt: "2 or minus 2")[$2 #h(0.2em) "or" #h(0.2em) − 2$] as algebraic solutions.], [], [#figure(figph[An image demonstrating the verification of p = -1 as a solution to an algebraic equation. The steps show substitution and simplification, confirming that both sides of the equation are equal to 2/3.], alt: "An image demonstrating the verification of p = -1 as a solution to an algebraic equation. The steps show substitution and simplification, confirming that both sides of the equation are equal to 2/3.", caption: none)], [], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve: #math.equation(block: false, alt: "the fraction 2 over x plus 1 plus the fraction 1 over x minus 1 equals the fraction 1 over x squared minus 1 .")[$frac(2, x + 1) + frac(1, x − 1) = frac(1, x^(2) − 1) .$] #solutionbox[ #math.equation(block: true, alt: "the fraction 2 over 3")[$frac(2, 3)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve: #math.equation(block: false, alt: "the fraction 5 over y plus 3 plus the fraction 2 over y minus 3 equals the fraction 5 over y squared minus 9 .")[$frac(5, y + 3) + frac(2, y − 3) = frac(5, y^(2) − 9) .$] #solutionbox[ #math.equation(block: true, alt: "2")[$2$] ] ] #examplebox("Example 5")[][ Solve: #math.equation(block: false, alt: "the fraction 4 over q minus 4 minus the fraction 3 over q minus 3 equals 1 .")[$frac(4, q − 4) − frac(3, q − 3) = 1 .$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[A mathematical equation is displayed, showing the expression 4/(q-4) - 3/(q-3) = 1. This algebraic equation involves rational terms with the variable 'q' in the denominators.], alt: "A mathematical equation is displayed, showing the expression 4/(q-4) - 3/(q-3) = 1. This algebraic equation involves rational terms with the variable 'q' in the denominators.", caption: none)]), [Note any value of the variable that would make any denominator zero.], [#figure(figph[A mathematical equation showing the sum of two fractions equaling one, with specified restrictions for the variable q. The equation is 4/(q-4) + 3/(q-3) = 1, where q is not equal to 4 or 3.], alt: "A mathematical equation showing the sum of two fractions equaling one, with specified restrictions for the variable q. The equation is 4/(q-4) + 3/(q-3) = 1, where q is not equal to 4 or 3.", caption: none)], [Find the least common denominator of all denominators in the equation. The LCD is#math.equation(block: false, alt: "open parenthesis q minus 4 close parenthesis open parenthesis q minus 3 close parenthesis")[$( q − 4 ) ( q − 3 )$].], [], [Clear the fractions by multiplying both sides of the equation by the LCD.], [#figure(figph[An algebraic equation showing a step where a rational expression is multiplied by the common denominator (q-4)(q-3) on both sides to clear the fractions.], alt: "An algebraic equation showing a step where a rational expression is multiplied by the common denominator (q-4)(q-3) on both sides to clear the fractions.", caption: none)], [Distribute.], [#figure(figph[An algebraic equation demonstrating the multiplication of terms by (q-4)(q-3) to clear denominators. Common factors are highlighted in red.], alt: "An algebraic equation demonstrating the multiplication of terms by (q-4)(q-3) to clear denominators. Common factors are highlighted in red.", caption: none)], [Remove common factors.], [#figure(figph[A mathematical equation illustrating the simplification of algebraic expressions by canceling common factors. The image shows (q-4) and (q-3) being canceled out in the numerators and denominators of the fractional terms.], alt: "A mathematical equation illustrating the simplification of algebraic expressions by canceling common factors. The image shows (q-4) and (q-3) being canceled out in the numerators and denominators of the fractional terms.", caption: none)], [Simplify.], [#figure(figph[An algebraic equation is displayed on a white background: 4(q-3) - 3(q-4) = (q-4)(q-3).], alt: "An algebraic equation is displayed on a white background: 4(q-3) - 3(q-4) = (q-4)(q-3).", caption: none)], [Simplify.], [#figure(figph[A mathematical equation is displayed on a white background: 4q - 12 - 3q + 12 = q^2 - 7q + 12. The equation appears to be part of an algebra problem, involving a variable 'q' and various constants.], alt: "A mathematical equation is displayed on a white background: 4q - 12 - 3q + 12 = q^2 - 7q + 12. The equation appears to be part of an algebra problem, involving a variable 'q' and various constants.", caption: none)], [Combine like terms.], [#figure(figph[A mathematical equation is displayed against a white background: q = q^2 - 7q + 12.], alt: "A mathematical equation is displayed against a white background: q = q^2 - 7q + 12.", caption: none)], [Solve. First write in standard form.], [#figure(figph[A quadratic equation is displayed against a white background, reading '0 = q^2 - 8q + 12' in black text.], alt: "A quadratic equation is displayed against a white background, reading '0 = q^2 - 8q + 12' in black text.", caption: none)], [Factor.], [#figure(figph[A mathematical equation on a white background, displaying 0 = (q - 2)(q - 6).], alt: "A mathematical equation on a white background, displaying 0 = (q - 2)(q - 6).", caption: none)], [Use the Zero Product Property.], [#figure(figph[The image shows the mathematical expression 'q = 2 or q = 6' in a simple, clear font on a white background.], alt: "The image shows the mathematical expression 'q = 2 or q = 6' in a simple, clear font on a white background.", caption: none)], [We did not get 4 or 3 as algebraic solutions.], [], [#figure(figph[This image demonstrates checking two potential solutions, q=2 and q=6, in the rational equation 4/(q-4) - 3/(q-3) = 1. Both values are substituted into the equation, and the calculations confirm that both q=2 and q=6 satisfy the equation, resulting in 1=1 for both cases.], alt: "This image demonstrates checking two potential solutions, q=2 and q=6, in the rational equation 4/(q-4) - 3/(q-3) = 1. Both values are substituted into the equation, and the calculations confirm that both q=2 and q=6 satisfy the equation, resulting in 1=1 for both cases.", caption: none)], [], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve: #math.equation(block: false, alt: "the fraction 2 over x plus 5 minus the fraction 1 over x minus 1 equals 1 .")[$frac(2, x + 5) − frac(1, x − 1) = 1 .$] #solutionbox[ #math.equation(block: true, alt: "−1 , −2")[$−1 , −2$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve: #math.equation(block: false, alt: "the fraction 3 over x plus 8 minus the fraction 2 over x minus 2 equals 1 .")[$frac(3, x + 8) − frac(2, x − 2) = 1 .$] #solutionbox[ #math.equation(block: true, alt: "−2 , −3")[$−2 , −3$] ] ] #examplebox("Example 6")[][ Solve: #math.equation(block: false, alt: "the fraction m plus 11 over m squared minus 5 m plus 4 equals the fraction 5 over m minus 4 minus the fraction 3 over m minus 1 .")[$frac(m + 11, m^(2) − 5 m + 4) = frac(5, m − 4) − frac(3, m − 1) .$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[An algebraic equation showing three equal rational expressions involving the variable 'm', for solving its value.], alt: "An algebraic equation showing three equal rational expressions involving the variable 'm', for solving its value.", caption: none)]), [Factor all the denominators, so we can note any value of the variable the would make any denominator zero.], [#figure(figph[The image displays a complex algebraic fraction (m+11)/((m-4)(m-1)) rewritten as the difference of two simpler fractions: 5/(m-4) - 3/(m-1), with restrictions m!=4 and m!=1.], alt: "The image displays a complex algebraic fraction (m+11)/((m-4)(m-1)) rewritten as the difference of two simpler fractions: 5/(m-4) - 3/(m-1), with restrictions m!=4 and m!=1.", caption: none)], [Find the least common denominator of all denominators in the equation. The LCD is#math.equation(block: false, alt: "open parenthesis m minus 4 close parenthesis open parenthesis m minus 1 close parenthesis")[$( m − 4 ) ( m − 1 )$].], [], [Clear the fractions.], [#figure(figph[An algebraic equation showing both sides multiplied by (m-4)(m-1) to clear denominators, simplifying the expression involving fractions with variables m.], alt: "An algebraic equation showing both sides multiplied by (m-4)(m-1) to clear denominators, simplifying the expression involving fractions with variables m.", caption: none)], [Distribute.], [#figure(figph[An algebraic equation demonstrates multiplying both sides by (m-4)(m-1) to clear denominators, simplifying the rational expression (m+11)/((m-4)(m-1)) and the terms 5/(m-4) and 3/(m-1).], alt: "An algebraic equation demonstrates multiplying both sides by (m-4)(m-1) to clear denominators, simplifying the rational expression (m+11)/((m-4)(m-1)) and the terms 5/(m-4) and 3/(m-1).", caption: none)], [Remove common factors.], [#figure(figph[An algebraic equation demonstrates the cancellation of common factors in red strike-through text to simplify both sides of the equation. It shows a step in solving for 'm'.], alt: "An algebraic equation demonstrates the cancellation of common factors in red strike-through text to simplify both sides of the equation. It shows a step in solving for 'm'.", caption: none)], [Simplify.], [#figure(figph[A mathematical equation is displayed against a white background: m + 11 = 5(m - 1) - 3(m - 4).], alt: "A mathematical equation is displayed against a white background: m + 11 = 5(m - 1) - 3(m - 4).", caption: none)], [Solve the resulting equation.], [#figure(figph[A mathematical equation is displayed: m + 11 = 5m - 5 - 3m + 12. The equation is presented in a clear, digital font on a white background.], alt: "A mathematical equation is displayed: m + 11 = 5m - 5 - 3m + 12. The equation is presented in a clear, digital font on a white background.", caption: none)], [], [#figure(figph[The image shows a simple mathematical equation in the center, which reads '4 = m', implying that the variable 'm' is equal to the number 4.], alt: "The image shows a simple mathematical equation in the center, which reads '4 = m', implying that the variable 'm' is equal to the number 4.", caption: none)], [Check. The only algebraic solution was 4, but we said that 4 would make a denominator equal to zero. The algebraic solution is an extraneous solution. There is no solution to this equation.], [], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve: #math.equation(block: false, alt: "the fraction x plus 13 over x squared minus 7 x plus 10 equals the fraction 6 over x minus 5 minus the fraction 4 over x minus 2 .")[$frac(x + 13, x^(2) − 7 x + 10) = frac(6, x − 5) − frac(4, x − 2) .$] #solutionbox[ no solution ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve: #math.equation(block: false, alt: "the fraction y minus 14 over y squared plus 3 y minus 4 equals the fraction 2 over y plus 4 plus the fraction 7 over y minus 1 .")[$frac(y − 14, y^(2) + 3 y − 4) = frac(2, y + 4) + frac(7, y − 1) .$] #solutionbox[ #math.equation(block: true, alt: "y equals minus 5")[$y = − 5$] ] ] The equation we solved in had only one algebraic solution, but it was an extraneous solution. That left us with no solution to the equation. Some equations have no solution. #examplebox("Example 7")[][ Solve: #math.equation(block: false, alt: "the fraction n over 12 plus the fraction n plus 3 over 3 n equals the fraction 1 over n .")[$frac(n, 12) + frac(n + 3, 3 n) = frac(1, n) .$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[A mathematical equation is displayed on a white background. The equation is n/12 + (n+3)/(3n) = 1/n, featuring fractions with variables in both numerators and denominators.], alt: "A mathematical equation is displayed on a white background. The equation is n/12 + (n+3)/(3n) = 1/n, featuring fractions with variables in both numerators and denominators.", caption: none)]), [Note any value of the variable that would make any denominator zero.], [#figure(figph[A mathematical equation is displayed: n/12 + (n+3)/(3n) = 1/n, with the condition that n is not equal to 0.], alt: "A mathematical equation is displayed: n/12 + (n+3)/(3n) = 1/n, with the condition that n is not equal to 0.", caption: none)], [Find the least common denominator of all denominators in the equation. The LCD is#math.equation(block: false, alt: "12 n")[$12 n$].], [], [Clear the fractions by multiplying both sides of the equation by the LCD.], [#figure(figph[A mathematical equation is shown with 12n multiplying a sum of fractions (n/12 + (n+3)/(3n)) on the left, and 12n multiplying a fraction (1/n) on the right, all in black text with '12n' in red.], alt: "A mathematical equation is shown with 12n multiplying a sum of fractions (n/12 + (n+3)/(3n)) on the left, and 12n multiplying a fraction (1/n) on the right, all in black text with '12n' in red.", caption: none)], [Distribute.], [#figure(figph[A mathematical equation is displayed, showing 12n multiplied by n/12, plus 12n multiplied by (n+3)/3n, equaling 12n multiplied by 1/n. The 12n terms are in red.], alt: "A mathematical equation is displayed, showing 12n multiplied by n/12, plus 12n multiplied by (n+3)/3n, equaling 12n multiplied by 1/n. The 12n terms are in red.", caption: none)], [Remove common factors.], [#figure(figph[A mathematical equation with red numbers and variables indicating terms being canceled out or simplified. The equation shows 12n(n/2) + 4 \* 3n((n+3)/3n) = 12n(1/n), with some denominators crossed out.], alt: "A mathematical equation with red numbers and variables indicating terms being canceled out or simplified. The equation shows 12n(n/2) + 4 * 3n((n+3)/3n) = 12n(1/n), with some denominators crossed out.", caption: none)], [Simplify.], [#figure(figph[A mathematical equation is displayed, showing 'n \* n + 4(n + 3) = 12 \* 1'.], alt: "A mathematical equation is displayed, showing 'n * n + 4(n + 3) = 12 * 1'.", caption: none)], [Solve the resulting equation.], [#figure(figph[A mathematical equation is displayed, reading 'n squared plus 4n plus 12 equals 12'.], alt: "A mathematical equation is displayed, reading 'n squared plus 4n plus 12 equals 12'.", caption: none)], [], [#figure(figph[A mathematical equation is displayed against a white background: r^2 + 4r = 0. The equation appears to be a quadratic equation in terms of the variable 'r'.], alt: "A mathematical equation is displayed against a white background: r^2 + 4r = 0. The equation appears to be a quadratic equation in terms of the variable 'r'.", caption: none)], [], [#figure(figph[A mathematical equation is displayed, showing n(n+4) = 0 on a white background, which is a common form for solving quadratic equations.], alt: "A mathematical equation is displayed, showing n(n+4) = 0 on a white background, which is a common form for solving quadratic equations.", caption: none)], [], [#figure(figph[The image displays the algebraic solution 'n = 0 or n = -4' in a simple, clear text format against a white background.], alt: "The image displays the algebraic solution 'n = 0 or n = -4' in a simple, clear text format against a white background.", caption: none)], [Check.], [], [#math.equation(block: false, alt: "n equals 0")[$n = 0$] is an extraneous solution.], [], [#figure(figph[Checking the solution n=-4 in an algebraic equation. The image shows the substitution, simplification, and verification that both sides of the equation are equal, confirming n=-4 is correct.], alt: "Checking the solution n=-4 in an algebraic equation. The image shows the substitution, simplification, and verification that both sides of the equation are equal, confirming n=-4 is correct.", caption: none)], [], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve: #math.equation(block: false, alt: "the fraction x over 18 plus the fraction x plus 6 over 9 x equals the fraction 2 over 3 x .")[$frac(x, 18) + frac(x + 6, 9 x) = frac(2, 3 x) .$] #solutionbox[ #math.equation(block: true, alt: "−2")[$−2$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve: #math.equation(block: false, alt: "the fraction y plus 5 over 5 y plus the fraction y over 15 equals the fraction 1 over y .")[$frac(y + 5, 5 y) + frac(y, 15) = frac(1, y) .$] #solutionbox[ #math.equation(block: true, alt: "−3")[$−3$] ] ] #examplebox("Example 8")[][ Solve: #math.equation(block: false, alt: "the fraction y over y plus 6 equals the fraction 72 over y squared minus 36 plus 4 .")[$frac(y, y + 6) = frac(72, y^(2) − 36) + 4 .$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[An algebraic equation is shown where y divided by the quantity y plus 6 equals 72 divided by the quantity y squared minus 36, plus 4.], alt: "An algebraic equation is shown where y divided by the quantity y plus 6 equals 72 divided by the quantity y squared minus 36, plus 4.", caption: none)]), [Factor all the denominators, so we can note any value of the variable that would make any denominator zero.], [#figure(figph[A mathematical equation is displayed: y / (y + 6) = 72 / ((y - 6)(y + 6)) + 4, with the restrictions y not equal to 6 and y not equal to -6.], alt: "A mathematical equation is displayed: y / (y + 6) = 72 / ((y - 6)(y + 6)) + 4, with the restrictions y not equal to 6 and y not equal to -6.", caption: none)], [Find the least common denominator. The LCD is#math.equation(block: false, alt: "open parenthesis y minus 6 close parenthesis open parenthesis y plus 6 close parenthesis")[$( y − 6 ) ( y + 6 )$].], [], [Clear the fractions.], [#figure(figph[A mathematical equation demonstrating the process of clearing fractions by multiplying both sides of the equation by the least common denominator, (y-6)(y+6).], alt: "A mathematical equation demonstrating the process of clearing fractions by multiplying both sides of the equation by the least common denominator, (y-6)(y+6).", caption: none)], [Simplify.], [#figure(figph[A mathematical equation is displayed on a white background: (y-6) \* y = 72 + (y-6)(y+6) \* 4. The characters are rendered in a black sans-serif font.], alt: "A mathematical equation is displayed on a white background: (y-6) * y = 72 + (y-6)(y+6) * 4. The characters are rendered in a black sans-serif font.", caption: none)], [Simplify.], [#figure(figph[A mathematical equation is shown: y(y-6) = 72 + 4(y^2 - 36). The equation features variables, numbers, parentheses, and operations, presented in a standard algebraic format on a white background.], alt: "A mathematical equation is shown: y(y-6) = 72 + 4(y^2 - 36). The equation features variables, numbers, parentheses, and operations, presented in a standard algebraic format on a white background.", caption: none)], [Solve the resulting equation.], [#figure(figph[An algebraic equation is shown: y^2 - 6y = 72 + 4y^2 - 144.], alt: "An algebraic equation is shown: y^2 - 6y = 72 + 4y^2 - 144.", caption: none)], [], [#figure(figph[A mathematical equation is displayed: 0 = 3y^2 + 6y - 72. It's a quadratic equation in the variable 'y' set equal to zero.], alt: "A mathematical equation is displayed: 0 = 3y^2 + 6y - 72. It's a quadratic equation in the variable 'y' set equal to zero.", caption: none)], [], [#figure(figph[A mathematical equation is shown with the expression 0 = 3(y^2 + 2y - 24).], alt: "A mathematical equation is shown with the expression 0 = 3(y^2 + 2y - 24).", caption: none)], [], [#figure(figph[A mathematical equation is displayed on a white background: 0 = 3(y + 6)(y - 4).], alt: "A mathematical equation is displayed on a white background: 0 = 3(y + 6)(y - 4).", caption: none)], [], [#figure(figph[The image displays mathematical equations, specifically 'y=-6, y=4' written in a clean, legible font against a plain white background.], alt: "The image displays mathematical equations, specifically 'y=-6, y=4' written in a clean, legible font against a plain white background.", caption: none)], [Check.], [], [#math.equation(block: false, alt: "y equals minus 6")[$y = − 6$] is an extraneous solution.], [], [#figure(figph[The image shows a step-by-step verification that y=4 is a valid solution for the equation y/(y+6) = 72/(y^2-36) + 4, demonstrating both sides equate to 4/10 after substitution and simplification.], alt: "The image shows a step-by-step verification that y=4 is a valid solution for the equation y/(y+6) = 72/(y^2-36) + 4, demonstrating both sides equate to 4/10 after substitution and simplification.", caption: none)], [], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve: #math.equation(block: false, alt: "the fraction x over x plus 4 equals the fraction 32 over x squared minus 16 plus 5 .")[$frac(x, x + 4) = frac(32, x^(2) − 16) + 5 .$] #solutionbox[ #math.equation(block: true, alt: "3")[$3$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve: #math.equation(block: false, alt: "the fraction y over y plus 8 equals the fraction 128 over y squared minus 64 plus 9 .")[$frac(y, y + 8) = frac(128, y^(2) − 64) + 9 .$] #solutionbox[ #math.equation(block: true, alt: "7")[$7$] ] ] #examplebox("Example 9")[][ Solve: #math.equation(block: false, alt: "the fraction x over 2 x minus 2 minus the fraction 2 over 3 x plus 3 equals the fraction 5 x squared minus 2 x plus 9 over 12 x squared minus 12 .")[$frac(x, 2 x − 2) − frac(2, 3 x + 3) = frac(5 x^(2) − 2 x + 9, 12 x^(2) − 12) .$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[A mathematical equation featuring rational expressions. The equation is x/(2x-2) - 2/(3x+3) = (5x^2 - 2x + 9)/(12x^2 - 12), with terms arranged horizontally.], alt: "A mathematical equation featuring rational expressions. The equation is x/(2x-2) - 2/(3x+3) = (5x^2 - 2x + 9)/(12x^2 - 12), with terms arranged horizontally.", caption: none)]), [We will start by factoring all denominators, to make it easier to identify extraneous solutions and the LCD.], [#figure(figph[A mathematical equation shows the subtraction of two algebraic fractions set equal to a third algebraic fraction. The equation is x over 2(x-1) minus 2 over 3(x+1) equals (5x^2 - 2x + 9) over 12(x-1)(x+1).], alt: "A mathematical equation shows the subtraction of two algebraic fractions set equal to a third algebraic fraction. The equation is x over 2(x-1) minus 2 over 3(x+1) equals (5x^2 - 2x + 9) over 12(x-1)(x+1).", caption: none)], [Note any value of the variable that would make any denominator zero.], [#figure(figph[A mathematical equation featuring fractions: x/(2(x-1)) - 2/(3(x+1)) = (5x^2 - 2x + 9)/(12(x-1)(x+1)), with the conditions x ≠ 1 and x ≠ -1.], alt: "A mathematical equation featuring fractions: x/(2(x-1)) - 2/(3(x+1)) = (5x^2 - 2x + 9)/(12(x-1)(x+1)), with the conditions x ≠ 1 and x ≠ -1.", caption: none)], [Find the least common denominator.The LCD is #math.equation(block: false, alt: "12 open parenthesis x minus 1 close parenthesis open parenthesis x plus 1 close parenthesis")[$12 ( x − 1 ) ( x + 1 )$]], [], [Clear the fractions.], [#figure(figph[An algebraic equation showing the multiplication of both sides by 12(x-1)(x+1) to eliminate fractional terms, a common step in solving rational equations.], alt: "An algebraic equation showing the multiplication of both sides by 12(x-1)(x+1) to eliminate fractional terms, a common step in solving rational equations.", caption: none)], [Simplify.], [#figure(figph[A mathematical equation is displayed, reading 6(x+1) \* x - 4(x-1) \* 2 = 5x^2 - 2x + 9. The equation involves algebraic expressions with variables and constants.], alt: "A mathematical equation is displayed, reading 6(x+1) * x - 4(x-1) * 2 = 5x^2 - 2x + 9. The equation involves algebraic expressions with variables and constants.", caption: none)], [Simplify.], [#figure(figph[A mathematical equation is displayed: 6x(x+1) - 4 \* 2(x-1) = 5x^2 - 2x + 9. It involves algebraic expressions with variables, constants, multiplication, subtraction, and equality.], alt: "A mathematical equation is displayed: 6x(x+1) - 4 * 2(x-1) = 5x^2 - 2x + 9. It involves algebraic expressions with variables, constants, multiplication, subtraction, and equality.", caption: none)], [Solve the resulting equation.], [#figure(figph[Quadratic equation: six x squared plus six x minus eight x plus eight equals five x squared minus two x plus nine.], alt: "Quadratic equation: six x squared plus six x minus eight x plus eight equals five x squared minus two x plus nine.", caption: none)], [], [#figure(figph[A mathematical equation on a white background reads 'x^2 - 1 = 0' in black text.], alt: "A mathematical equation on a white background reads 'x^2 - 1 = 0' in black text.", caption: none)], [], [#figure(figph[An image displays the algebraic equation (x-1)(x+1)=0, a common form of a quadratic equation.], alt: "An image displays the algebraic equation (x-1)(x+1)=0, a common form of a quadratic equation.", caption: none)], [], [#figure(figph[A mathematical equation is displayed on a white background, stating 'x = 1 or x = -1'.], alt: "A mathematical equation is displayed on a white background, stating 'x = 1 or x = -1'.", caption: none)], [Check.], [], [#math.equation(block: false, alt: "x equals 1")[$x = 1$] and #math.equation(block: false, alt: "x equals minus 1")[$x = − 1$] are extraneous solutions. #linebreak() The equation has no solution.], [], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve: #math.equation(block: false, alt: "the fraction y over 5 y minus 10 minus the fraction 5 over 3 y plus 6 equals the fraction 2 y squared minus 19 y plus 54 over 15 y squared minus 60 .")[$frac(y, 5 y − 10) − frac(5, 3 y + 6) = frac(2 y^(2) − 19 y + 54, 15 y^(2) − 60) .$] #solutionbox[ no solution ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve: #math.equation(block: false, alt: "the fraction z over 2 z plus 8 minus the fraction 3 over 4 z minus 8 equals the fraction 3 z squared minus 16 z minus 16 over 8 z squared plus 16 z minus 64 .")[$frac(z, 2 z + 8) − frac(3, 4 z − 8) = frac(3 z^(2) − 16 z − 16, 8 z^(2) + 16 z − 64) .$] #solutionbox[ no solution ] ] === Solve a Rational Equation for a Specific Variable When we solved linear equations, we learned how to solve a formula for a specific variable. Many formulas used in business, science, economics, and other fields use rational equations to model the relation between two or more variables. We will now see how to solve a rational equation for a specific variable. We’ll start with a formula relating distance, rate, and time. We have used it many times before, but not usually in this form. #examplebox("Example 10")[][ Solve: #math.equation(block: false, alt: "the fraction D over T equals R for T .")[$frac(D, T) = R #h(0.2em) "for" #h(0.2em) T .$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[The image displays the equation D/T = R for T, representing a mathematical relationship where D divided by T equals R, with the explicit instruction to solve or express the equation for T.], alt: "The image displays the equation D/T = R for T, representing a mathematical relationship where D divided by T equals R, with the explicit instruction to solve or express the equation for T.", caption: none)]), [Note any value of the variable that would make any denominator zero.], [#figure(figph[A mathematical equation displays D over T equals R, with the condition that T is not equal to 0, representing a division relationship with a non-zero denominator.], alt: "A mathematical equation displays D over T equals R, with the condition that T is not equal to 0, representing a division relationship with a non-zero denominator.", caption: none)], [Clear the fractions by multiplying both sides of the equations by the LCD, #emph[T].], [#figure(figph[A mathematical equation T(D/T) = T(R) is displayed. The initial 'T' and the 'T' after the equals sign are red, while the other characters are black.], alt: "A mathematical equation T(D/T) = T(R) is displayed. The initial 'T' and the 'T' after the equals sign are red, while the other characters are black.", caption: none)], [Simplify.], [#figure(figph[The mathematical formula D=T\*R is displayed, representing the relationship between Distance, Time, and Rate. The letters are bold and in a simple font on a white background.], alt: "The mathematical formula D=T*R is displayed, representing the relationship between Distance, Time, and Rate. The letters are bold and in a simple font on a white background.", caption: none)], [Divide both sides by #emph[R] to isolate #emph[T].], [#figure(figph[A mathematical equation shows D/R = RT/R, with the variable R in the denominator of both fractions highlighted in red, indicating a step in solving or manipulating the equation.], alt: "A mathematical equation shows D/R = RT/R, with the variable R in the denominator of both fractions highlighted in red, indicating a step in solving or manipulating the equation.", caption: none)], [Simplify.], [#figure(figph[A mathematical formula is displayed: D divided by R equals T. This represents the relationship where Time (T) is equal to Distance (D) divided by Rate (R).], alt: "A mathematical formula is displayed: D divided by R equals T. This represents the relationship where Time (T) is equal to Distance (D) divided by Rate (R).", caption: none)], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve: #math.equation(block: false, alt: "the fraction A over L equals W")[$frac(A, L) = W$] for #math.equation(block: false, alt: "L .")[$L .$] #solutionbox[ #math.equation(block: true, alt: "L equals the fraction A over W")[$L = frac(A, W)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve: #math.equation(block: false, alt: "the fraction F over A equals M")[$frac(F, A) = M$] for #math.equation(block: false, alt: "A .")[$A .$] #solutionbox[ #math.equation(block: true, alt: "A equals the fraction F over M")[$A = frac(F, M)$] ] ] uses the formula for slope that we used to get the point-slope form of an equation of a line. #examplebox("Example 11")[][ Solve: #math.equation(block: false, alt: "m equals the fraction x minus 2 over y minus 3 for y .")[$m = frac(x − 2, y − 3) #h(0.2em) "for" #h(0.2em) y .$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[An algebraic problem displaying the equation m = (x - 2) / (y - 3) with instructions to solve for y.], alt: "An algebraic problem displaying the equation m = (x - 2) / (y - 3) with instructions to solve for y.", caption: none)]), [Note any value of the variable that would make any denominator zero.], [#figure(figph[A mathematical equation displays the slope m as a fraction: m = (x-2) / (y-3), with the condition that y is not equal to 3.], alt: "A mathematical equation displays the slope m as a fraction: m = (x-2) / (y-3), with the condition that y is not equal to 3.", caption: none)], [Clear the fractions by multiplying both sides of the equations by the LCD, #math.equation(block: false, alt: "y minus 3")[$y − 3$].], [#figure(figph[The equation (y-3)m = (y-3) \* ((x-2)/(y-3)) is shown, with the common factor (y-3) highlighted in red.], alt: "The equation (y-3)m = (y-3) * ((x-2)/(y-3)) is shown, with the common factor (y-3) highlighted in red.", caption: none)], [Simplify.], [#figure(figph[A mathematical equation is displayed against a white background, reading 'ym - 3m = x - 2' in black text.], alt: "A mathematical equation is displayed against a white background, reading 'ym - 3m = x - 2' in black text.", caption: none)], [Isolate the term with #emph[y].], [#figure(figph[A mathematical equation is displayed on a white background, which reads 'ym = x - 2 + 3m' in black text.], alt: "A mathematical equation is displayed on a white background, which reads 'ym = x - 2 + 3m' in black text.", caption: none)], [Divide both sides by #emph[m] to isolate #emph[y].], [#figure(figph[A mathematical equation shows 'ym over m equals x minus 2 plus 3m over m' with the variable 'm' in the denominator highlighted in red. The numerator of the first fraction is 'ym', and the denominator is 'm'. The numerator of the second fraction is 'x - 2 + 3m', and the denominator is 'm'.], alt: "A mathematical equation shows 'ym over m equals x minus 2 plus 3m over m' with the variable 'm' in the denominator highlighted in red. The numerator of the first fraction is 'ym', and the denominator is 'm'. The numerator of the second fraction is 'x - 2 + 3m', and the denominator is 'm'.", caption: none)], [Simplify.], [#figure(figph[A mathematical equation shows 'y = (x - 2 + 3m) / m' presented on a white background.], alt: "A mathematical equation shows 'y = (x - 2 + 3m) / m' presented on a white background.", caption: none)], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve: #math.equation(block: false, alt: "the fraction y minus 2 over x plus 1 equals the fraction 2 over 3")[$frac(y − 2, x + 1) = frac(2, 3)$] for #math.equation(block: false, alt: "x .")[$x .$] #solutionbox[ #math.equation(block: true, alt: "x equals the fraction 3 y minus 8 over 2")[$x = frac(3 y − 8, 2)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve: #math.equation(block: false, alt: "x equals the fraction y over 1 minus y")[$x = frac(y, 1 − y)$] for #math.equation(block: false, alt: "y .")[$y .$] #solutionbox[ #math.equation(block: true, alt: "y equals the fraction x over 1 plus x")[$y = frac(x, 1 + x)$] ] ] Be sure to follow all the steps in . It may look like a very simple formula, but we cannot solve it instantly for either denominator. #examplebox("Example 12")[][ Solve #math.equation(block: false, alt: "the fraction 1 over c plus the fraction 1 over m equals 1 for c .")[$frac(1, c) + frac(1, m) = 1 #h(0.2em) "for" #h(0.2em) c .$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[A mathematical equation showing the sum of two fractions, 1 over c plus 1 over m, equaling 1, with the instruction to solve for c.], alt: "A mathematical equation showing the sum of two fractions, 1 over c plus 1 over m, equaling 1, with the instruction to solve for c.", caption: none)]), [Note any value of the variable that would make any denominator zero.], [#figure(figph[A mathematical equation showing the sum of two reciprocals equal to one: 1/c + 1/m = 1, with the conditions that c is not equal to 0 and m is not equal to 0.], alt: "A mathematical equation showing the sum of two reciprocals equal to one: 1/c + 1/m = 1, with the conditions that c is not equal to 0 and m is not equal to 0.", caption: none)], [Clear the fractions by multiplying both sides of the equations by the LCD, #math.equation(block: false, alt: "c m")[$c m$].], [#figure(figph[A mathematical equation shows 'cm multiplied by the sum of 1 over c and 1 over m' equals 'cm multiplied by 1'. The letters 'cm' are in red, while the rest of the equation is in black.], alt: "A mathematical equation shows 'cm multiplied by the sum of 1 over c and 1 over m' equals 'cm multiplied by 1'. The letters 'cm' are in red, while the rest of the equation is in black.", caption: none)], [Distribute.], [#figure(figph[A mathematical equation on a white background, displaying cm(1/c) + cm(1/m) = cm(1), with 'cm' in red and the rest in black.], alt: "A mathematical equation on a white background, displaying cm(1/c) + cm(1/m) = cm(1), with 'cm' in red and the rest in black.", caption: none)], [Simplify.], [#figure(figph[A simple mathematical equation is displayed on a white background, reading 'm + C = cm' in black text.], alt: "A simple mathematical equation is displayed on a white background, reading 'm + C = cm' in black text.", caption: none)], [Collect the terms with #emph[c] to the right.], [#figure(figph[A mathematical equation on a white background, displaying 'm = cm - c'.], alt: "A mathematical equation on a white background, displaying 'm = cm - c'.", caption: none)], [Factor the expression on the right.], [#figure(figph[A mathematical equation is displayed on a white background, which reads 'm = c(m - 1)'.], alt: "A mathematical equation is displayed on a white background, which reads 'm = c(m - 1)'.", caption: none)], [To isolate #emph[c], divide both sides by #math.equation(block: false, alt: "m minus 1")[$m − 1$].], [#figure(figph[A mathematical equation shown as m/(m-1) = c(m-1)/(m-1). The denominator (m-1) is highlighted in red on both sides of the equation.], alt: "A mathematical equation shown as m/(m-1) = c(m-1)/(m-1). The denominator (m-1) is highlighted in red on both sides of the equation.", caption: none)], [Simplify by removing common factors.], [#figure(figph[A mathematical equation showing 'm' divided by 'm minus 1' equals 'c'.], alt: "A mathematical equation showing 'm' divided by 'm minus 1' equals 'c'.", caption: none)], )) Notice that even though we excluded #math.equation(block: false, alt: "c equals 0 and m equals 0")[$c = 0 #h(0.2em) "and" #h(0.2em) m = 0$] from the original equation, we must also now state that #math.equation(block: false, alt: "m not equal to 1")[$m ≠ 1$]. ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve: #math.equation(block: false, alt: "the fraction 1 over a plus the fraction 1 over b equals c")[$frac(1, a) + frac(1, b) = c$] for #math.equation(block: false, alt: "a .")[$a .$] #solutionbox[ #math.equation(block: true, alt: "a equals the fraction b over c b minus 1")[$a = frac(b, c b − 1)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve: #math.equation(block: false, alt: "the fraction 2 over x plus the fraction 1 over 3 equals the fraction 1 over y")[$frac(2, x) + frac(1, 3) = frac(1, y)$] for #math.equation(block: false, alt: "y .")[$y .$] #solutionbox[ #math.equation(block: true, alt: "y equals the fraction 3 x over 6 plus x")[$y = frac(3 x, 6 + x)$] ] ] === Key Concepts - #strong[Strategy to Solve Equations with Rational Expressions] + Note any value of the variable that would make any denominator zero. + Find the least common denominator of #emph[all] denominators in the equation. + Clear the fractions by multiplying both sides of the equation by the LCD. + Solve the resulting equation. + Check. - If any values found in Step 1 are algebraic solutions, discard them. - Check any remaining solutions in the original equation. ==== Practice Makes Perfect #strong[Solve Rational Equations] In the following exercises, solve. #math.equation(block: true, alt: "the fraction 1 over a plus the fraction 2 over 5 equals the fraction 1 over 2")[$frac(1, a) + frac(2, 5) = frac(1, 2)$] #solutionbox[ #math.equation(block: true, alt: "10")[$10$] ] #math.equation(block: true, alt: "the fraction 5 over 6 plus the fraction 3 over b equals the fraction 1 over 3")[$frac(5, 6) + frac(3, b) = frac(1, 3)$] #math.equation(block: true, alt: "the fraction 5 over 2 minus the fraction 1 over c equals the fraction 3 over 4")[$frac(5, 2) − frac(1, c) = frac(3, 4)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 4 over 7")[$frac(4, 7)$] ] #math.equation(block: true, alt: "the fraction 6 over 3 minus the fraction 2 over d equals the fraction 4 over 9")[$frac(6, 3) − frac(2, d) = frac(4, 9)$] #math.equation(block: true, alt: "the fraction 4 over 5 plus the fraction 1 over 4 equals the fraction 2 over v")[$frac(4, 5) + frac(1, 4) = frac(2, v)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 40 over 21")[$frac(40, 21)$] ] #math.equation(block: true, alt: "the fraction 3 over 7 plus the fraction 2 over 3 equals the fraction 1 over w")[$frac(3, 7) + frac(2, 3) = frac(1, w)$] #math.equation(block: true, alt: "the fraction 7 over 9 plus the fraction 1 over x equals the fraction 2 over 3")[$frac(7, 9) + frac(1, x) = frac(2, 3)$] #solutionbox[ #math.equation(block: true, alt: "−9")[$−9$] ] #math.equation(block: true, alt: "the fraction 3 over 8 plus the fraction 2 over y equals the fraction 1 over 4")[$frac(3, 8) + frac(2, y) = frac(1, 4)$] #math.equation(block: true, alt: "1 minus the fraction 2 over m equals the fraction 8 over m squared")[$1 − frac(2, m) = frac(8, m^(2))$] #solutionbox[ #math.equation(block: true, alt: "−2 , 4")[$−2 , 4$] ] #math.equation(block: true, alt: "1 plus the fraction 4 over n equals the fraction 21 over n squared")[$1 + frac(4, n) = frac(21, n^(2))$] #math.equation(block: true, alt: "1 plus the fraction 9 over p equals the fraction −20 over p squared")[$1 + frac(9, p) = frac(−20, p^(2))$] #solutionbox[ #math.equation(block: true, alt: "−5 , −4")[$−5 , −4$] ] #math.equation(block: true, alt: "1 minus the fraction 7 over q equals the fraction −6 over q squared")[$1 − frac(7, q) = frac(−6, q^(2))$] #math.equation(block: true, alt: "the fraction 1 over r plus 3 equals the fraction 4 over 2 r")[$frac(1, r + 3) = frac(4, 2 r)$] #solutionbox[ #math.equation(block: true, alt: "−6")[$−6$] ] #math.equation(block: true, alt: "the fraction 3 over t minus 6 equals the fraction 1 over t")[$frac(3, t − 6) = frac(1, t)$] #math.equation(block: true, alt: "the fraction 5 over 3 v minus 2 equals the fraction 7 over 4 v")[$frac(5, 3 v − 2) = frac(7, 4 v)$] #solutionbox[ #math.equation(block: true, alt: "14")[$14$] ] #math.equation(block: true, alt: "the fraction 8 over 2 w plus 1 equals the fraction 3 over w")[$frac(8, 2 w + 1) = frac(3, w)$] #math.equation(block: true, alt: "the fraction 3 over x plus 4 plus the fraction 7 over x minus 4 equals the fraction 8 over x squared minus 16")[$frac(3, x + 4) + frac(7, x − 4) = frac(8, x^(2) − 16)$] #solutionbox[ #math.equation(block: true, alt: "the fraction −4 over 5")[$frac(−4, 5)$] ] #math.equation(block: true, alt: "the fraction 5 over y minus 9 plus the fraction 1 over y plus 9 equals the fraction 18 over y squared minus 81")[$frac(5, y − 9) + frac(1, y + 9) = frac(18, y^(2) − 81)$] #math.equation(block: true, alt: "the fraction 8 over z minus 10 plus the fraction 7 over z plus 10 equals the fraction 5 over z squared minus 100")[$frac(8, z − 10) + frac(7, z + 10) = frac(5, z^(2) − 100)$] #solutionbox[ #math.equation(block: true, alt: "minus the fraction 1 over 3")[$− frac(1, 3)$] ] #math.equation(block: true, alt: "the fraction 9 over a plus 11 plus the fraction 6 over a minus 11 equals the fraction 7 over a squared minus 121")[$frac(9, a + 11) + frac(6, a − 11) = frac(7, a^(2) − 121)$] #math.equation(block: true, alt: "the fraction 1 over q plus 4 minus the fraction 2 over q minus 2 equals 1")[$frac(1, q + 4) − frac(2, q − 2) = 1$] #solutionbox[ #math.equation(block: true, alt: "minus 2 , minus 1")[$− 2 , #h(0.2em) − 1$] ] #math.equation(block: true, alt: "the fraction 3 over r plus 10 minus the fraction 4 over r minus 4 equals 1")[$frac(3, r + 10) − frac(4, r − 4) = 1$] #math.equation(block: true, alt: "the fraction 1 over t plus 7 minus the fraction 5 over t minus 5 equals 1")[$frac(1, t + 7) − frac(5, t − 5) = 1$] #solutionbox[ #math.equation(block: true, alt: "−5 , −1")[$−5 , −1$] ] #math.equation(block: true, alt: "the fraction 2 over s plus 7 minus the fraction 3 over s minus 3 equals 1")[$frac(2, s + 7) − frac(3, s − 3) = 1$] #math.equation(block: true, alt: "the fraction v minus 10 over v squared minus 5 v plus 4 equals the fraction 3 over v minus 1 minus the fraction 6 over v minus 4")[$frac(v − 10, v^(2) − 5 v + 4) = frac(3, v − 1) − frac(6, v − 4)$] #solutionbox[ #math.equation(block: true, alt: "no solution")[$"no solution"$] ] #math.equation(block: true, alt: "the fraction w plus 8 over w squared minus 11 w plus 28 equals the fraction 5 over w minus 7 plus the fraction 2 over w minus 4")[$frac(w + 8, w^(2) − 11 w + 28) = frac(5, w − 7) + frac(2, w − 4)$] #math.equation(block: true, alt: "the fraction x minus 10 over x squared plus 8 x plus 12 equals the fraction 3 over x plus 2 plus the fraction 4 over x plus 6")[$frac(x − 10, x^(2) + 8 x + 12) = frac(3, x + 2) + frac(4, x + 6)$] #solutionbox[ #math.equation(block: true, alt: "no solution")[$"no solution"$] ] #math.equation(block: true, alt: "the fraction y minus 3 over y squared minus 4 y minus 5 equals the fraction 1 over y plus 1 plus the fraction 8 over y minus 5")[$frac(y − 3, y^(2) − 4 y − 5) = frac(1, y + 1) + frac(8, y − 5)$] #math.equation(block: true, alt: "the fraction z over 16 plus the fraction z plus 2 over 4 z equals the fraction 1 over 2 z")[$frac(z, 16) + frac(z + 2, 4 z) = frac(1, 2 z)$] #solutionbox[ #math.equation(block: true, alt: "−4")[$−4$] ] #math.equation(block: true, alt: "the fraction a over 9 plus the fraction a plus 3 over 3 a equals the fraction 1 over a")[$frac(a, 9) + frac(a + 3, 3 a) = frac(1, a)$] #math.equation(block: true, alt: "the fraction b plus 3 over 3 b plus the fraction b over 24 equals the fraction 1 over b")[$frac(b + 3, 3 b) + frac(b, 24) = frac(1, b)$] #solutionbox[ #math.equation(block: true, alt: "−8")[$−8$] ] #math.equation(block: true, alt: "the fraction c plus 3 over 12 c plus the fraction c over 36 equals the fraction 1 over 4 c")[$frac(c + 3, 12 c) + frac(c, 36) = frac(1, 4 c)$] #math.equation(block: true, alt: "the fraction d over d plus 3 equals the fraction 18 over d squared minus 9 plus 4")[$frac(d, d + 3) = frac(18, d^(2) − 9) + 4$] #solutionbox[ #math.equation(block: true, alt: "2")[$2$] ] #math.equation(block: true, alt: "the fraction m over m plus 5 equals the fraction 50 over m squared minus 25 plus 6")[$frac(m, m + 5) = frac(50, m^(2) − 25) + 6$] #math.equation(block: true, alt: "the fraction n over n plus 2 equals the fraction 8 over n squared minus 4 plus 3")[$frac(n, n + 2) = frac(8, n^(2) − 4) + 3$] #solutionbox[ #math.equation(block: true, alt: "1")[$1$] ] #math.equation(block: true, alt: "the fraction p over p plus 7 equals the fraction 98 over p squared minus 49 plus 8")[$frac(p, p + 7) = frac(98, p^(2) − 49) + 8$] #math.equation(block: true, alt: "the fraction q over 3 q minus 9 minus the fraction 3 over 4 q plus 12")[$frac(q, 3 q − 9) − frac(3, 4 q + 12)$] #linebreak() #math.equation(block: true, alt: "equals the fraction 7 q squared plus 6 q plus 63 over 24 q squared minus 216")[$#h(1.5em) = frac(7 q^(2) + 6 q + 63, 24 q^(2) − 216)$] #solutionbox[ #math.equation(block: true, alt: "no solution")[$"no solution"$] ] #math.equation(block: true, alt: "the fraction r over 3 r minus 15 minus the fraction 1 over 4 r plus 20")[$frac(r, 3 r − 15) − frac(1, 4 r + 20)$] #linebreak() #math.equation(block: true, alt: "equals the fraction 3 r squared plus 17 r plus 40 over 12 r squared minus 300")[$#h(1.5em) = frac(3 r^(2) + 17 r + 40, 12 r^(2) − 300)$] #math.equation(block: true, alt: "the fraction s over 2 s plus 6 minus the fraction 2 over 5 s plus 5")[$frac(s, 2 s + 6) − frac(2, 5 s + 5)$] #linebreak() #math.equation(block: true, alt: "equals the fraction 5 s squared minus s minus 18 over 10 s squared plus 40 s plus 30")[$#h(1.5em) = frac(5 s^(2) − s − 18, 10 s^(2) + 40 s + 30)$] #solutionbox[ #math.equation(block: true, alt: "no solution")[$"no solution"$] ] #math.equation(block: true, alt: "the fraction t over 6 t minus 12 minus the fraction 5 over 2 t plus 10")[$frac(t, 6 t − 12) − frac(5, 2 t + 10)$] #linebreak() #math.equation(block: true, alt: "equals the fraction t squared minus 23 t plus 70 over 12 t squared plus 36 t minus 120")[$#h(1.5em) = frac(t^(2) − 23 t + 70, 12 t^(2) + 36 t − 120)$] #strong[Solve a Rational Equation for a Specific Variable] In the following exercises, solve. #math.equation(block: true, alt: "the fraction C over r equals 2 π for r")[$frac(C, r) = 2 π #h(0.2em) "for" #h(0.2em) r$] #solutionbox[ #math.equation(block: true, alt: "r equals the fraction C over 2 π")[$r = frac(C, 2 π)$] ] #math.equation(block: true, alt: "the fraction I over r equals P for r")[$frac(I, r) = P #h(0.2em) "for" #h(0.2em) r$] #math.equation(block: true, alt: "the fraction V over h equals l w for h")[$frac(V, h) = l w #h(0.2em) "for" #h(0.2em) h$] #solutionbox[ #math.equation(block: true, alt: "h equals the fraction v over l w")[$h = frac(v, l w)$] ] #math.equation(block: true, alt: "the fraction 2 A over b equals h for b")[$frac(2 A, b) = h #h(0.2em) "for" #h(0.2em) b$] #math.equation(block: true, alt: "the fraction v plus 3 over w minus 1 equals the fraction 1 over 2 for w")[$frac(v + 3, w − 1) = frac(1, 2) #h(0.2em) "for" #h(0.2em) w$] #solutionbox[ #math.equation(block: true, alt: "w equals 2 v plus 7")[$w = 2 v + 7$] ] #math.equation(block: true, alt: "the fraction x plus 5 over 2 minus y equals the fraction 4 over 3 for y")[$frac(x + 5, 2 − y) = frac(4, 3) #h(0.2em) "for" #h(0.2em) y$] #math.equation(block: true, alt: "a equals the fraction b plus 3 over c minus 2 for c")[$a = frac(b + 3, c − 2) #h(0.2em) "for" #h(0.2em) c$] #solutionbox[ #math.equation(block: true, alt: "c equals the fraction b plus 3 plus 2 a over a")[$c = frac(b + 3 + 2 a, a)$] ] #math.equation(block: true, alt: "m equals the fraction n over 2 minus n for n")[$m = frac(n, 2 − n) #h(0.2em) "for" #h(0.2em) n$] #math.equation(block: true, alt: "the fraction 1 over p plus the fraction 2 over q equals 4 for p")[$frac(1, p) + frac(2, q) = 4 #h(0.2em) "for" #h(0.2em) p$] #solutionbox[ #math.equation(block: true, alt: "p equals the fraction q over 4 q minus 2")[$p = frac(q, 4 q − 2)$] ] #math.equation(block: true, alt: "the fraction 3 over s plus the fraction 1 over t equals 2 for s")[$frac(3, s) + frac(1, t) = 2 #h(0.2em) "for" #h(0.2em) s$] #math.equation(block: true, alt: "the fraction 2 over v plus the fraction 1 over 5 equals the fraction 3 over w for w")[$frac(2, v) + frac(1, 5) = frac(3, w) #h(0.2em) "for" #h(0.2em) w$] #solutionbox[ #math.equation(block: true, alt: "w equals the fraction 15 v over 10 plus v")[$w = frac(15 v, 10 + v)$] ] #math.equation(block: true, alt: "the fraction 6 over x plus the fraction 2 over 3 equals the fraction 1 over y for y")[$frac(6, x) + frac(2, 3) = frac(1, y) #h(0.2em) "for" #h(0.2em) y$] #math.equation(block: true, alt: "the fraction m plus 3 over n minus 2 equals the fraction 4 over 5 for n")[$frac(m + 3, n − 2) = frac(4, 5) #h(0.2em) "for" #h(0.2em) n$] #solutionbox[ #math.equation(block: true, alt: "n equals the fraction 5 m plus 23 over 4")[$n = frac(5 m + 23, 4)$] ] #math.equation(block: true, alt: "the fraction E over c equals m squared for c")[$frac(E, c) = m^(2) #h(0.2em) "for" #h(0.2em) c$] #math.equation(block: true, alt: "the fraction 3 over x minus the fraction 5 over y equals the fraction 1 over 4 for y")[$frac(3, x) − frac(5, y) = frac(1, 4) #h(0.2em) "for" #h(0.2em) y$] #solutionbox[ #math.equation(block: true, alt: "y equals the fraction 20 x over 12 minus x")[$y = frac(20 x, 12 − x)$] ] #math.equation(block: true, alt: "the fraction R over T equals W for T")[$frac(R, T) = W #h(0.2em) "for" #h(0.2em) T$] #math.equation(block: true, alt: "r equals the fraction s over 3 minus t for t")[$r = frac(s, 3 − t) #h(0.2em) "for" #h(0.2em) t$] #solutionbox[ #math.equation(block: true, alt: "t equals the fraction 3 r minus s over r")[$t = frac(3 r − s, r)$] ] #math.equation(block: true, alt: "c equals the fraction 2 over a plus the fraction b over 5 for a")[$c = frac(2, a) + frac(b, 5) #h(0.2em) "for" #h(0.2em) a$] ==== Everyday Math #strong[House Painting] Alain can paint a house in 4 days. Spiro would take 7 days to paint the same house. Solve the equation #math.equation(block: false, alt: "the fraction 1 over 4 plus the fraction 1 over 7 equals the fraction 1 over t")[$frac(1, 4) + frac(1, 7) = frac(1, t)$] for #emph[t] to find the number of days it would take them to paint the house if they worked together. #solutionbox[ #math.equation(block: false, alt: "2 the fraction 6 over 11")[$2 frac(6, 11)$] days ] #strong[Boating] Ari can drive his boat 18 miles with the current in the same amount of time it takes to drive 10 miles against the current. If the speed of the boat is 7 knots, solve the equation #math.equation(block: false, alt: "the fraction 18 over 7 plus c equals the fraction 10 over 7 minus c")[$frac(18, 7 + c) = frac(10, 7 − c)$] for #emph[c] to find the speed of the current. ==== Writing Exercises Why is there no solution to the equation #math.equation(block: false, alt: "the fraction 3 over x minus 2 equals the fraction 5 over x minus 2")[$frac(3, x − 2) = frac(5, x − 2)$]? #solutionbox[ Answers will vary. ] Pete thinks the equation #math.equation(block: false, alt: "the fraction y over y plus 6 equals the fraction 72 over y squared minus 36 plus 4")[$frac(y, y + 6) = frac(72, y^(2) − 36) + 4$] has two solutions, #math.equation(block: false, alt: "y equals −6 and y equals 4")[$y = −6 #h(0.2em) "and" #h(0.2em) y = 4$]. Explain why Pete is wrong. ==== Self Check ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section. #figure(figph[This table has three rows and four columns. The first row is a header row and it labels each column. The first column is labeled "I can …", the second "Confidently", the third “With some help” and the last "No–I don’t get it". In the “I can…” column the next row reads “solve rational equations”. The next row reads, “solve rational equations for a specific variable”. The remaining columns are blank.], alt: "This table has three rows and four columns. The first row is a header row and it labels each column. The first column is labeled \"I can …\", the second \"Confidently\", the third “With some help” and the last \"No–I don’t get it\". In the “I can…” column the next row reads “solve rational equations”. The next row reads, “solve rational equations for a specific variable”. The remaining columns are blank.", caption: none) ⓑ After reviewing this checklist, what will you do to become confident for all objectives?