Elementary Algebra 2e · free preview

8.3 Add and Subtract Rational Expressions with a Common Denominator

Learning objectives: Add rational expressions with a common denominator; Subtract rational expressions with a common denominator; Add and subtract rational expressions whose denominators are opposites

1. Learning objectives

Learning objectives

Learning objectives

  • Add rational expressions with a common denominator
  • Subtract rational expressions with a common denominator
  • Add and subtract rational expressions whose denominators are opposites

2. Add Rational Expressions with a Common Denominator

Add Rational Expressions with a Common Denominator

What is the first step you take when you add numerical fractions? You check if they have a common denominator. If they do, you add the numerators and place the sum over the common denominator. If they do not have a common denominator, you find one before you add.

It is the same with rational expressions. To add rational expressions, they must have a common denominator. When the denominators are the same, you add the numerators and place the sum over the common denominator.

If p,q, and rp,q,\,\mathrm{and}\,r are polynomials where r≠0r\ne{}0, then

pr+qr=p+qr\frac{p}{r}+\frac{q}{r}=\frac{p+q}{r}

To add rational expressions with a common denominator, add the numerators and place the sum over the common denominator.

We will add two numerical fractions first, to remind us of how this is done.

3. Example 8.30

Example 8.30

Example 8.30

Add: 518+718.\frac{5}{18}+\frac{7}{18}.

Solution

518+718\frac{5}{18}+\frac{7}{18}
The fractions have a commondenominator, so add the numerators andplace the sum over the common denominator.5+718\frac{5+7}{18}
Add in the numerator.1218\frac{12}{18}
Factor the numerator and denominator toshow the common factors.6⋅26⋅3\frac{6\cdot{}2}{6\cdot{}3}
Remove common factors.6⋅26⋅3\frac{6\cdot{}2}{6\cdot{}3}
Simplify.23\frac{2}{3}

Remember, we do not allow values that would make the denominator zero. What value of yy should be excluded in the next example?

4. Example 8.31

Example 8.31

Example 8.31

Add: 3y4y−3+74y−3.\frac{3y}{4y-3}+\frac{7}{4y-3}.

Solution

3y4y−3+74y−3\frac{3y}{4y-3}+\frac{7}{4y-3}
The fractions have a commondenominator, so add the numerators andplace the sum over the common denominator.3y+74y−3\frac{3y+7}{4y-3}

The numerator and denominator cannot be factored. The fraction is simplified.

5. Example 8.32

Example 8.32

Example 8.32

Add: 7x+12x+3+x2x+3.\frac{7x+12}{x+3}+\frac{{x}^{2}}{x+3}.

Solution

7x+12x+3+x2x+3\frac{7x+12}{x+3}+\frac{{x}^{2}}{x+3}
The fractions have a commondenominator, so add the numerators andplace the sum over the common denominator.7x+12+x2x+3\frac{7x+12+{x}^{2}}{x+3}
Write the degrees in descending order.x2+7x+12x+3\frac{{x}^{2}+7x+12}{x+3}
Factor the numerator.(x+3)(x+4)x+3\frac{(x+3)(x+4)}{x+3}
Simplify by removing common factors.(x+3)(x+4)x+3\frac{(x+3)(x+4)}{x+3}
Simplify.x+4x+4

6. Subtract Rational Expressions with a Common Denominator

Subtract Rational Expressions with a Common Denominator

To subtract rational expressions, they must also have a common denominator. When the denominators are the same, you subtract the numerators and place the difference over the common denominator.

If p,q, and rp,q,\,\mathrm{and}\,r are polynomials where r≠0r\ne{}0, then

pr−qr=p−qr\frac{p}{r}-\frac{q}{r}=\frac{p-q}{r}

To subtract rational expressions, subtract the numerators and place the difference over the common denominator.

We always simplify rational expressions. Be sure to factor, if possible, after you subtract the numerators so you can identify any common factors.

7. Example 8.33

Example 8.33

Example 8.33

Subtract: n2n−10−100n−10.\frac{{n}^{2}}{n-10}-\frac{100}{n-10}.

Solution

n2n−10−100n−10\frac{{n}^{2}}{n-10}-\frac{100}{n-10}
The fractions have a commondenominator, so subtract the numeratorsand place the difference over the common denominator.n2−100n−10\frac{{n}^{2}-100}{n-10}
Factor the numerator.(n−10)(n+10)n−10\frac{(n-10)(n+10)}{n-10}
Simplify by removing common factors.(n−10)(n+10)n−10\frac{(n-10)(n+10)}{n-10}
Simplify.n+10n+10

Be careful of the signs when you subtract a binomial!

8. Example 8.34

Example 8.34

Example 8.34

Subtract: y2y−6−2y+24y−6.\frac{{y}^{2}}{y-6}-\frac{2y+24}{y-6}.

Solution

y2y−6−2y+24y−6\frac{{y}^{2}}{y-6}-\frac{2y+24}{y-6}
The fractions have a commondenominator, so subtract the numeratorsand place the difference over the common denominator.y2−(2y+24)y−6\frac{{y}^{2}-(2y+24)}{y-6}
Distribute the sign in the numerator.y2−2y−24y−6\frac{{y}^{2}-2y-24}{y-6}
Factor the numerator.(y−6)(y+4)y−6\frac{(y-6)(y+4)}{y-6}
Remove common factors.(y−6)(y+4)y−6\frac{(y-6)(y+4)}{y-6}
Simplify.y+4y+4

9. Example 8.35

Example 8.35

Example 8.35

Subtract: 5x2−7x+3x2−3x−18−4x2+x−9x2−3x−18.\frac{5{x}^{2}-7x+3}{{x}^{2}-3x-18}-\frac{4{x}^{2}+x-9}{{x}^{2}-3x-18}.

Solution

5x2−7x+3x2−3x−18−4x2+x−9x2−3x−18\frac{5{x}^{2}-7x+3}{{x}^{2}-3x-18}-\frac{4{x}^{2}+x-9}{{x}^{2}-3x-18}
Subtract the numerators and place thedifference over the common denominator.5x2−7x+3−(4x2+x−9)x2−3x−18\frac{5{x}^{2}-7x+3-(4{x}^{2}+x-9)}{{x}^{2}-3x-18}
Distribute the sign in the numerator.5x2−7x+3−4x2−x+9x2−3x−18\frac{5{x}^{2}-7x+3-4{x}^{2}-x+9}{{x}^{2}-3x-18}
Combine like terms.x2−8x+12x2−3x−18\frac{{x}^{2}-8x+12}{{x}^{2}-3x-18}
Factor the numerator and the denominator.(x−2)(x−6)(x+3)(x−6)\frac{(x-2)(x-6)}{(x+3)(x-6)}
Simplify by removing common factors.(x−2)(x−6)(x+3)(x−6)\frac{(x-2)(x-6)}{(x+3)(x-6)}
Simplify.(x−2)(x+3)\frac{(x-2)}{(x+3)}

10. Add and Subtract Rational Expressions whose Denominators are Opposites

Add and Subtract Rational Expressions whose Denominators are Opposites

When the denominators of two rational expressions are opposites, it is easy to get a common denominator. We just have to multiply one of the fractions by −1−1\frac{-1}{-1}.

Let’s see how this works.

11. Example 8.36

Example 8.36

Example 8.36

Add: 4u−13u−1+u1−3u.\frac{4u-1}{3u-1}+\frac{u}{1-3u}.

Solution

12. Example 8.37

Example 8.37

Example 8.37

Subtract: m2−6mm2−1−3m+21−m2.\frac{{m}^{2}-6m}{{m}^{2}-1}-\frac{3m+2}{1-{m}^{2}}.

Solution

13. Section 8.3 Exercises

Section 8.3 Exercises

Section 8.3 Exercises

Add Rational Expressions with a Common Denominator

In the following exercises, add.

Subtract Rational Expressions with a Common Denominator

In the following exercises, subtract.

Add and Subtract Rational Expressions whose Denominators are Opposites

In the following exercises, add.

In the following exercises, subtract.

Everyday Math

Writing Exercises

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

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

14. Practice 1

  1. Add: 716+516.\frac{7}{16}+\frac{5}{16}.
  1. Add: 310+110.\frac{3}{10}+\frac{1}{10}.
  1. Add: 5x2x+3+22x+3.\frac{5x}{2x+3}+\frac{2}{2x+3}.

15. Practice 2

  1. Add: xx−2+1x−2.\frac{x}{x-2}+\frac{1}{x-2}.
  1. Add: 9x+14x+7+x2x+7.\frac{9x+14}{x+7}+\frac{{x}^{2}}{x+7}.
  1. Add: x2+8xx+5+15x+5.\frac{{x}^{2}+8x}{x+5}+\frac{15}{x+5}.

16. Practice 3

  1. Subtract: x2x+3−9x+3.\frac{{x}^{2}}{x+3}-\frac{9}{x+3}.
  1. Subtract: 4x22x−5−252x−5.\frac{4{x}^{2}}{2x-5}-\frac{25}{2x-5}.
  1. Subtract: n2n−4−n+12n−4.\frac{{n}^{2}}{n-4}-\frac{n+12}{n-4}.

17. Practice 4

  1. Subtract: y2y−1−9y−8y−1.\frac{{y}^{2}}{y-1}-\frac{9y-8}{y-1}.
  1. Subtract: 4x2−11x+8x2−3x+2−3x2+x−3x2−3x+2.\frac{4{x}^{2}-11x+8}{{x}^{2}-3x+2}-\frac{3{x}^{2}+x-3}{{x}^{2}-3x+2}.
  1. Subtract: 6x2−x+20x2−81−5x2+11x−7x2−81.\frac{6{x}^{2}-x+20}{{x}^{2}-81}-\frac{5{x}^{2}+11x-7}{{x}^{2}-81}.

18. Practice 5

  1. Add: 8x−152x−5+2x5−2x.\frac{8x-15}{2x-5}+\frac{2x}{5-2x}.
  1. Add: 6y2+7y−104y−7+2y2+2y+117−4y.\frac{6{y}^{2}+7y-10}{4y-7}+\frac{2{y}^{2}+2y+11}{7-4y}.
  1. Subtract: y2−5yy2−4−6y−64−y2.\frac{{y}^{2}-5y}{{y}^{2}-4}-\frac{6y-6}{4-{y}^{2}}.

19. Practice 6

  1. Subtract: 2n2+8n−1n2−1−n2−7n−11−n2.\frac{2{n}^{2}+8n-1}{{n}^{2}-1}-\frac{{n}^{2}-7n-1}{1-{n}^{2}}.
  1. 215+715\frac{2}{15}+\frac{7}{15}
  1. 724+1124\frac{7}{24}+\frac{11}{24}

20. Practice 7

  1. 3aa−b+1a−b\frac{3a}{a-b}+\frac{1}{a-b}
  1. dd+8+5d+8\frac{d}{d+8}+\frac{5}{d+8}
  1. p2+10pp+2+16p+2\frac{{p}^{2}+10p}{p+2}+\frac{16}{p+2}

21. Practice 8

  1. 2r22r−1+15r−82r−1\frac{2{r}^{2}}{2r-1}+\frac{15r-8}{2r-1}
  1. 8t2t+4+32tt+4\frac{8{t}^{2}}{t+4}+\frac{32t}{t+4}
  1. 2w2w2−16+8ww2−16\frac{2{w}^{2}}{{w}^{2}-16}+\frac{8w}{{w}^{2}-16}

22. Practice 9

  1. y2y+8−64y+8\frac{{y}^{2}}{y+8}-\frac{64}{y+8}
  1. 9a23a−7−493a−7\frac{9{a}^{2}}{3a-7}-\frac{49}{3a-7}
  1. c2c−8−6c+16c−8\frac{{c}^{2}}{c-8}-\frac{6c+16}{c-8}

23. Practice 10

  1. 3m26m−30−21m−306m−30\frac{3{m}^{2}}{6m-30}-\frac{21m-30}{6m-30}
  1. 6p2+3p+4p2+4p−5−5p2+p+7p2+4p−5\frac{6{p}^{2}+3p+4}{{p}^{2}+4p-5}-\frac{5{p}^{2}+p+7}{{p}^{2}+4p-5}
  1. 5r2+7r−33r2−49−4r2+5r+30r2−49\frac{5{r}^{2}+7r-33}{{r}^{2}-49}-\frac{4{r}^{2}+5r+30}{{r}^{2}-49}

24. Practice 11

  1. 10v2v−1+2v+41−2v\frac{10v}{2v-1}+\frac{2v+4}{1-2v}
  1. 10x2+16x−78x−3+2x2+3x−13−8x\frac{10{x}^{2}+16x-7}{8x-3}+\frac{2{x}^{2}+3x-1}{3-8x}
  1. z2+6zz2−25−3z+2025−z2\frac{{z}^{2}+6z}{{z}^{2}-25}-\frac{3z+20}{25-{z}^{2}}

25. Practice 12

  1. 2b2+30b−13b2−49−2b2−5b−849−b2\frac{2{b}^{2}+30b-13}{{b}^{2}-49}-\frac{2{b}^{2}-5b-8}{49-{b}^{2}}
  1. 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 8r\frac{8}{r} and her biking time is modeled by the expression 24r+4.\frac{24}{r+4}. Add the rational expressions 8r+24r+4\frac{8}{r}+\frac{24}{r+4} to get an expression for the total amount of time Sarah ran and biked.
  1. If Pete can paint a wall in pp hours, then in one hour he can paint 1p\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 1p+3\frac{1}{p+3} of the wall. Add the rational expressions 1p+1p+3\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.

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