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Mathematics LibreTexts

7.1E: Exercises

  • Page ID
    30950
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    Practice Makes Perfect

    In the following exercises, determine the values for which the rational expression is undefined.

    Example \(\PageIndex{49}\)

    1. \(\frac{2x}{z}\)
    2. \(\frac{4p−1}{6p−5}\)
    3. \(\frac{n−3}{n^2+2n−8}\)
    Answer
    1. z=0
    2. \(p=\frac{5}{6}\)
    3. n=−4, n=2

    Example \(\PageIndex{50}\)

    1. \(\frac{10m}{11n}\)
    2. \(\frac{6y+13}{4y−9}\)
    3. \(\frac{b−8}{b^2−36}\)

    Example \(\PageIndex{51}\)

    1. \(\frac{4x^{2}y}{3y}\)
    2. \(\frac{3x−2}{2x+1}\)
    3. \(\frac{u−1}{u^2−3u−28}\)
    Answer
    1. y=0
    2. \(x=−\frac{1}{2}\)
    3. u=−4, u=7

    Example \(\PageIndex{52}\)

    1. \(\frac{5pq^{2}}{9q}\)
    2. \(\frac{7a−4}{3a+5}\)
    3. \(\frac{1}{x^2−4}\)
    Evaluate Rational Expressions

    In the following exercises, evaluate the rational expression for the given values.

    Example \(\PageIndex{53}\)

    \(\frac{2x}{x−1}\)

    1. x=0
    2. x=2
    3. x=−1
    Answer
    1. 0
    2. 4
    3. 1

    Example \(\PageIndex{54}\)

    \(\frac{4y−1}{5y−3}\)

    1. y=0
    2. y=2
    3. y=−1

    Example \(\PageIndex{55}\)

    \(\frac{2p+3}{p^2+1}\)

    1. p=0
    2. p=1
    3. p=−2
    Answer
    1. 3
    2. \(\frac{5}{2}\)
    3. \(−\frac{1}{5}\)

    Example \(\PageIndex{56}\)

    \(\frac{x+3}{2−3x}\)

    1. x=0
    2. x=1
    3. x=−2

    Example \(\PageIndex{57}\)

    \(\frac{y^2+5y+6}{y^2−1}\)

    1. y=0
    2. y=2
    3. y=−2
    Answer
    1. −6
    2. \(\frac{20}{3}\)
    3. 0

    Example \(\PageIndex{58}\)

    \(\frac{z^2+3z−10}{z^2−1}\)

    1. z=0
    2. z=2
    3. z=−2

    Example \(\PageIndex{59}\)

    \(\frac{a^2−4}{a^2+5a+4}\)

    1. a=0
    2. a=1
    3. a=−2
    Answer
    1. −1
    2. \(−\frac{3}{10}\)
    3. 0

    Example \(\PageIndex{60}\)

    \(\frac{b^2+2}{b^2−3b−4}\)

    1. b=0
    2. b=2
    3. b=−2

    Example \(\PageIndex{61}\)

    \(\frac{x^2+3xy+2y^2}{2x^{3}y}\)

    1. x=1, y=−1
    2. x=2, y=1
    3. x=−1, y=−2
    Answer
    1. 0
    2. \(\frac{3}{4}\)
    3. \(\frac{15}{4}\)

    Example \(\PageIndex{62}\)

    \(\frac{c^2+cd−2d^2}{cd^{3}}\)

    1. c=2, d=−1
    2. c=1, d=−1
    3. c=−1, d=2

    Example \(\PageIndex{63}\)

    \(\frac{m^2−4n^2}{5mn^3}\)

    1. m=2, n=1
    2. m=−1, n=−1
    3. m=3, n=2
    Answer
    1. 0
    2. \(−\frac{3}{5}\)
    3. \(−\frac{7}{20}\)

    Example \(\PageIndex{64}\)

    \(\frac{2s^{2}t}{s^2−9t^2}\)

    1. s=4, t=1
    2. s=−1, t=−1
    3. s=0, t=2
    ​​​​​​​Simplify Rational Expressions

    In the following exercises, simplify.

    Example \(\PageIndex{65}\)

    \(−\frac{4}{52}\)

    Answer

    \(−\frac{1}{13}\)

    Example \(\PageIndex{66}\)

    \(−\frac{44}{55}\)

    Example \(\PageIndex{67}\)

    \(\frac{56}{63}\)

    Answer

    \(\frac{8}{9}\)

    Example \(\PageIndex{68}\)

    \(\frac{65}{104}\)

    Example \(\PageIndex{69}\)

    \(\frac{6ab^{2}}{12a^{2}b}\)

    Answer

    \(\frac{b}{2ab}\)

    Example \(\PageIndex{70}\)

    \(\frac{15xy^{3}}{x^{3}y^{3}}\)

    Example \(\PageIndex{71}\)

    \(\frac{8m^{3}n}{12mn^2}\)

    Answer

    \(\frac{2m^2}{3n}\)

    Example \(\PageIndex{72}\)

    \(\frac{36v^{3}w^2}{27vw^3}\)

    Example \(\PageIndex{73}\)

    \(\frac{3a+6}{4a+8}\)

    Answer

    \(\frac{3}{4}\)

    Example \(\PageIndex{74}\)

    \(\frac{5b+5}{6b+6}\)

    Example \(\PageIndex{75}\)

    \(\frac{3c−9}{5c−15}\)

    Answer

    \(\frac{3}{5}\)

    Example \(\PageIndex{76}\)

    \(\frac{4d+8}{9d+18}\)

    Example \(\PageIndex{77}\)

    \(\frac{7m+63}{5m+45}\)

    Answer

    \(\frac{7}{5}\)

    Example \(\PageIndex{78}\)

    \(\frac{8n−96}{3n−36}\)

    Exercise \(\PageIndex{79}\)

    \(\frac{12p−240}{5p−100}\)

    Answer

    \(\frac{12}{5}\)

    Example \(\PageIndex{80}\)

    \(\frac{6q+210}{5q+175}\)

    Example \(\PageIndex{81}\)

    \(\frac{a^2−a−12}{a^2−8a+16}\)

    Answer

    \(\frac{a+3}{a−4}\)

    Example \(\PageIndex{82}\)

    \(\frac{x^2+4x−5}{x^2−2x+1}\)

    Example \(\PageIndex{83}\)

    \(\frac{y^2+3y−4}{y^2−6y+5}\)

    Answer

    \(\frac{y+4}{y−5}\)

    Example \(\PageIndex{84}\)

    \(\frac{v^2+8v+15}{v^2−v−12}\)

    Example \(\PageIndex{85}\)

    \(\frac{x^2−25}{x^2+2x−15}\)

    Answer

    \(\frac{x−5}{x−3}\)

    Example \(\PageIndex{86}\)

    \(\frac{a^2−4}{a^2+6a−16}\)

    Example \(\PageIndex{87}\)

    \(\frac{y^2−2y−3}{y^2−9}\)

    Answer

    \(\frac{y+1}{y+3}\)

    Example \(\PageIndex{88}\)

    \(\frac{b^2+9b+18}{b^2−36}\)

    Example \(\PageIndex{89}\)

    \(\frac{y^3+y^2+y+1}{y^2+2y+1}\)

    Answer

    \(\frac{y^2+1}{y+1}\)​​​​​​​

    Example \(\PageIndex{90}\)

    \(\frac{p^3+3p^2+4p+12}{p^2+p−6}\)

    Example \(\PageIndex{91}\)

    \(\frac{x^3−2x^2−25x+50}{x^2−25}\)

    Answer

    x−2

    Example \(\PageIndex{92}\)

    \(\frac{q^3+3q^2−4q−12}{q^2−4}\)

    Example \(\PageIndex{93}\)

    \(\frac{3a^2+15a}{6a^2+6a−36}\)

    Answer

    \(\frac{a(a+5)}{2(a+3)(a−2)}\)

    Example \(\PageIndex{94}\)

    \(\frac{8b^2−32b}{2b^2−6b−80}\)

    Example \(\PageIndex{95}\)

    \(\frac{−5c^2−10c}{−10c^2+30c+100}\)

    Answer

    \(\frac{c}{2(c−5)}\)

    Example \(\PageIndex{96}\)

    \(\frac{4d^2−24d}{2d^2−4d−48}\)

    Example \(\PageIndex{97}\)

    \(\frac{3m^2+30m+75}{4m^2−100}\)

    Answer

    \(\frac{3(m+5)}{4(m−5)}\)

    Example \(\PageIndex{98}\)

    \(\frac{5n^2+30n+45}{2n^2−18}\)

    Example \(\PageIndex{99}\)

    \(\frac{5r^2+30r−35}{r^2−49}\)

    Answer

    \(\frac{5(r−1)}{r+7}\)

    Example \(\PageIndex{100}\)

    \(\frac{3s^2+30s+72}{3s^2−48}\)

    Example \(\PageIndex{101}\)

    \(\frac{t^3−27}{t^2−9}\)​​​​​​​

    Answer

    \(\frac{t^2+3t+9}{t+3}\)

    Example \(\PageIndex{102}\)

    \(\frac{v^3−1}{v^2−1}\)

    Example \(\PageIndex{103}\)

    \(\frac{w^3+216}{w^2−36}\)

    Answer

    \(\frac{w^2−6w+36}{w−6}\)

    Example \(\PageIndex{104}\)

    \(\frac{v^3+125}{v^2−25}\)

    Simplify Rational Expressions with Opposite Factors

    In the following exercises, simplify each rational expression.

    Example \(\PageIndex{105}\)

    \(\frac{a−5}{5−a}\)

    Answer

    −1

    Example \(\PageIndex{106}\)

    \(\frac{b−12}{12−b}\)

    Example \(\PageIndex{107}\)

    \(\frac{11−c}{c−11}\)

    Answer

    −1

    Example \(\PageIndex{108}\)

    \(\frac{5−d}{d−5}\)

    Example \(\PageIndex{109}\)

    \(\frac{12−2x}{x^2−36}\)

    Answer

    \(−\frac{2}{x+6}\)

    Example \(\PageIndex{110}\)

    \(\frac{20−5y}{y^2−16}\)

    Example \(\PageIndex{111}\)

    \(\frac{4v−32}{64−v^2}\)

    Answer

    \(−\frac{4}{8+v}\)

    Example \(\PageIndex{112}\)

    \(\frac{7w−21}{9−w^2}\)

    Example \(\PageIndex{113}\)

    \(\frac{y^2−11y+24}{9−y^2}\)

    Answer

    \(−\frac{y−8}{3+y}\)

    Example \(\PageIndex{114}\)

    \(\frac{z^2−9z+20}{16−z^2}\)

    Example \(\PageIndex{115}\)

    \(\frac{a^2−5a−36}{81−a^2}\)

    Answer

    \(−\frac{a+4}{9+a}\)​​​​​​​

    Example \(\PageIndex{116}\)

    \(\frac{b^2+b−42}{36−b^2}\)​​​​​​​

    Everyday Math

    Example \(\PageIndex{117}\)

    Tax Rates For the tax year 2015, the amount of tax owed by a single person earning between $37,450 and $90,750, can be found by evaluating the formula 0.25x−4206.25, where x is income. The average tax rate for this income can be found by evaluating the formula \(\frac{0.25x−4206.25}{x}\). What would be the average tax rate for a single person earning $50,000?

    Answer

    16.5%

    Example \(\PageIndex{118}\)

    Work The length of time it takes for two people for perform the same task if they work together can be found by evaluating the formula \(\frac{xy}{x+y}\). If Tom can paint the den in x=45 minutes and his brother Bobby can paint it in y=60 minutes, how many minutes will it take them if they work together?

    Writing Exercises

    Example \(\PageIndex{119}\)

    Explain how you find the values of x for which the rational expression \(\frac{x^2−x−20}{x^2−4}\) is undefined.​​​​​​​

    Answer

    Answers will vary, but all should reference setting the denominator function to zero.

    Example \(\PageIndex{120}\)

    Explain all the steps you take to simplify the rational expression \(\frac{p^2+4p−21}{9−p^2}\).​​​​​​​

    Self Check

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

    This figure shows a table with four columns and five rows. The first row is a header row and each column is labeled. The first column header is labeled “I can…”, the second is labeled “Confidently”, the third is labeled “With some help”, and the fourth is labeled “No—I don’t get it!” In the first column under “I can”, the cells read “determine the values for which a rational expression is undefined,” “evaluate rational expressions,” “simplify rational expressions,” and “simplify rational expressions with opposite factors.” The rest of the cells are blank.

    ⓑ If most of your checks were:

    …confidently. Congratulations! You have achieved your goals 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 as topics you do not master become potholes in your road to success. Math is sequential - every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Who 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 critical and you must not ignore it. You need to get help immediately or you will quickly be overwhelmed. See your instructor as soon as possible to discuss your situation. Together you can come up with a plan to get you the help you need.

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