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7.3E: Exersices

  • Page ID
    79498
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    Practice Makes Perfect

    Add Rational Expressions with a Common Denominator

    In the following exercises, add.

    Example \(\PageIndex{25}\)

    \(\frac{2}{15}+\frac{7}{15}\)

    Answer

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

    Example \(\PageIndex{26}\)

    \(\frac{4}{21}+\frac{3}{21}\)

    Example \(\PageIndex{27}\)

    \(\frac{7}{24}+\frac{11}{24}\)

    Answer

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

    Example \(\PageIndex{28}\)

    \(\frac{7}{36}+\frac{13}{36}\)

    Example \(\PageIndex{29}\)

    \(\frac{3a}{a−b}+\frac{1}{a−b}\)

    Answer

    \(\frac{3a+1}{a-b}\)

    Example \(\PageIndex{30}\)

    \(\frac{3c}{4c−5}+\frac{5}{4c−5}\)

    Example \(\PageIndex{31}\)

    \(\frac{d}{d+8}+\frac{5}{d+8}\)

    Answer

    \(\frac{d+5}{d+8}\)

    Example \(\PageIndex{32}\)

    \(\frac{7m}{2m+n}+\frac{4}{2m+n}\)

    Example \(\PageIndex{33}\)

    \(\frac{p^2+10p}{p+2}+\frac{16}{p+2}\)

    Answer

    p+8

    Example \(\PageIndex{34}\)

    \(\frac{q^2+12q}{q+3}+\frac{27}{q+3}\)

    Example \(\PageIndex{35}\)

    \(\frac{2r^2}{2r−1}+\frac{15r−8}{2r−1}\)

    Answer

    r+8

    Example \(\PageIndex{36}\)

    \(\frac{3s^2}{3s−2}+\frac{13s−10}{3s−2}\)

    Example \(\PageIndex{37}\)

    \(\frac{8t^2}{t+4}+\frac{32t}{t+4}\)

    Answer

    8t

    Example \(\PageIndex{38}\)

    \(\frac{6v^2}{v+5}+\frac{30v}{v+5}\)

    Example \(\PageIndex{39}\)

    \(\frac{2w^2}{w^2−16}+\frac{8w}{w^2−16}\)

    Answer

    \(\frac{2w}{w−4}\)

    Example \(\PageIndex{40}\)

    \(\frac{7x^2}{x^2−9}+\frac{21x}{x^2−9}\)

    ​​​​​​​Subtract Rational Expressions with a Common Denominator

    In the following exercises, subtract.

    Example \(\PageIndex{41}\)

    \(\frac{y^2}{y+8}−\frac{64}{y+8}\)

    Answer

    y−8​​​​​​​

    Example \(\PageIndex{42}\)

    \(\frac{z^2}{z+2}−\frac{4}{z+2}\)​​​​​​​

    Example \(\PageIndex{43}\)

    \(\frac{9a^2}{3a−7}−\frac{49}{3a−7}\)

    Answer

    3a+7

    Example \(\PageIndex{44}\)

    \(\frac{25b^2}{5b−6}−\frac{36}{5b−6}\)

    Example \(\PageIndex{45}\)

    \(\frac{c^2}{c−8}−\frac{6c+16}{c−8}\)

    Answer

    c+2

    Example \(\PageIndex{46}\)

    \(\frac{d^2}{d−9}−\frac{6d+27}{d−9}\)

    Example \(\PageIndex{47}\)

    \(\frac{3m^2}{6m−30}−\frac{21m−30}{6m−30}\)

    Answer

    \(\frac{m−2}{3}\)

    Example \(\PageIndex{48}\)

    \(\frac{2n^2}{4n−32}−\frac{30n−16}{4n−32}\)

    Example \(\PageIndex{49}\)

    \(\frac{6p^2+3p+4}{p^2+4p−5}−\frac{5p^2+p+7}{p^2+4p−5}\)

    Answer

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

    Example \(\PageIndex{50}\)

    \(\frac{5q^2+3q−9}{q^2+6q+8}−\frac{4q^2+9q+7}{q^2+6q+8}\)

    Example \(\PageIndex{51}\)

    \(\frac{5r^2+7r−33}{r^2−49}−\frac{4r^2−5r−30}{r^2−49}\)

    Answer

    \(\frac{r+9}{r+7}\)

    Example \(\PageIndex{52}\)

    \(\frac{7t^2−t−4}{t^2−25}−\frac{6t^2+2t−1}{t^2−25}\)

    ​​​​​​​Add and Subtract Rational Expressions whose Denominators are Opposites

    In the following exercises, add.

    Example \(\PageIndex{53}\)

    \(\frac{10v^2}{v−1}+\frac{2v+4}{1−2v}\)

    Answer

    4​​​​​​​

    Example \(\PageIndex{54}\)

    \(\frac{20w}{5w−2}+\frac{5w+6}{2−5w}\)

    Example \(\PageIndex{55}\)

    \(\frac{10x^2+16x−7}{8x−3}+\frac{2x^2+3x−1}{3−8x}\)

    Answer

    x+2

    Example \(\PageIndex{56}\)

    \(\frac{6y^2+2y−11}{3y−7}+\frac{3y^2−3y+17}{7−3y}\)

    ​​​​​​​In the following exercises, subtract.

    Example \(\PageIndex{57}\)

    \(\frac{z^2+6z}{z^2−25}−\frac{3z+20}{25−z^2}\)

    Answer

    \(\frac{z+4}{z−5}\)

    Example \(\PageIndex{58}\)

    \(\frac{a^2+3a}{a^2−9}−\frac{3a−27}{9−a^2}\)

    Example \(\PageIndex{59}\)

    \(\frac{2b^2+30b−13}{b^2−49}−\frac{2b^2−5b−8}{49−b^2}\)

    Answer

    \(\frac{4b−3}{b−7}\)

    Example \(\PageIndex{60}\)

    \(\frac{c^2+5c−10}{c^2−16}−\frac{c^2−8c−10}{16−c^2}\)​​​​​​​

    Everyday Math

    Example \(\PageIndex{61}\)

    Sarah ran 8 miles and then biked 24 miles. Her biking speed is 4 mph faster than her running speed. If rr represents Sarah’s speed when she ran, then her running time is modeled by the expression \(\frac{8}{r}\) and her biking time is modeled by the expression \(\frac{24}{r+4}\). Add the rational expressions \(\frac{8}{r}+\frac{24}{r+4}\) to get an expression for the total amount of time Sarah ran and biked.

    Answer

    \(\frac{32r+32}{r(r+4)}\)

    Example \(\PageIndex{62}\)

    If Pete can paint a wall in pp hours, then in one hour he can paint \(\frac{1}{p}\) of the wall. It would take Penelope 3 hours longer than Pete to paint the wall, so in one hour she can paint \(\frac{1}{p+3}\) of the wall. Add the rational expressions \(\frac{1}{p}+\frac{1}{p+3}\) to get an expression for the part of the wall Pete and Penelope would paint in one hour if they worked together.

    Writing Exercises

    Example \(\PageIndex{63}\)

    Donald thinks that \(\frac{3}{x}+\frac{4}{x}\) is \(\frac{7}{2x}\). Is Donald correct? Explain.

    Example \(\PageIndex{64}\)

    Explain how you find the Least Common Denominator of \(x^2+5x+4\) and \(x^2−16\).​​​​​​​

    Self Check

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

    The above image is a table with four columns and four rows. The first row is the header row. The first header is labeled “I can…”, the second “Confidently”, the third, “With some help”, and the fourth “No – I don’t get it!”. In the first column under “I can”, the next row reads “add rational expressions with a common denominator.”, the next row reads “subtract rational expressions with a common denominator.”, the next row reads, “add and subtract rational expressions whose denominators are opposites.”, the last row reads “What does this checklist tell you about your mastery of this section? What steps will you take to improve?” The remaining columns are blank.

    ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?


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