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Section1.9Adding and subtracting rational expressions

Subsection1.9.1Written Examples

1Combining rational expressions that have common denominators

Recall that when adding or subtracting fractions that have a common denominator, we add or subtract the numerators over a single occurrence of the denominator. We then simplify the result (i.e. reduce the resultant fraction — if it reduces). Two examples are shown below.

\begin{align*} \frac{1}{9}+\frac{5}{9}\amp=\frac{1+5}{9}\\ \amp=\frac{6}{9}\\ \amp=\frac{2}{3} \end{align*}

\begin{align*} \frac{4}{15}-\frac{7}{15}\amp=\frac{4-7}{15}\\ \amp=\frac{-3}{15}\\ \amp=-\frac{1}{5} \end{align*}

The same strategy is used when adding or subtracting rational expressions that have a common denominator. Several examples follow.

Example

\begin{align*} \frac{x+3}{x-7}+\frac{x-17}{x-7}\amp=\frac{x+3+x-17}{x-7}\\ \amp=\frac{2x-14}{x-7}\\ \amp=\frac{2(x-7)}{x-7}\\ \amp=\frac{x-7}{x-7} \cdot \frac{2}{1}\\ \amp=2, x \neq 7 \end{align*}
2Combining rational expressions that have denominators with no common factors

3Combining rational expressions that have denominators with both common factors and non-common factors

4Combining rational expressions that have denominators that include opposite factors

Subsection1.9.2Practice Exercises

1Combining rational expressions that have common denominators

Add and/or subtract each set of expressions. Make sure that you simplify each result.

  1. \(\frac{2x}{x-7}-\frac{14}{x-7}\)
  2. \(\frac{x^2}{x^2+4x+4}+\frac{9x+14}{x^2+4x+4}\)
  3. \(\frac{2x^2}{x^2-6x-72}-\frac{240-4x}{x^2-6x-72}\)

Solution

  1. \(\begin{aligned}[t] \frac{2x}{x-7}-\frac{14}{x-7}\amp=\frac{2x-14}{x-7}\\ \amp=\frac{2(x-7)}{x-7}\\ \amp=2, x \neq 7 \end{aligned}\) \begin{equation*} \end{equation*}
  2. \(\begin{aligned}[t] \frac{x^2}{x^2+4x+4}+\frac{9x+14}{x^2+4x+4}\amp=\frac{x^2+9x+14}{x^2+4x+4}\\ \amp=\frac{(x+7)(x+2)}{(x+2)(x+2)}\\ \amp=\frac{x+2}{x+2} \cdot \frac{x+7}{x+2}\\ \amp=\frac{x+7}{x+2} \end{aligned}\) \begin{equation*} \end{equation*}
  3. \(\begin{aligned}[t] \frac{2x^2}{x^2-6x-72}-\frac{240-4x}{x^2-6x-72}\amp=\frac{2x^2-240+4x}{x^2-6x-72}\\ \amp=\frac{2(x^2+2x-120)}{x^2-6x-72}\\ \amp=\frac{2(x+12)(x-10)}{(x+12)(x-6)}\\ \amp=\frac{x+12}{x+12} \cdot \frac{2(x-10)}{x-6}\\ \amp=\frac{2(x-10)}{x-6}, x \neq 12 \end{aligned}\)
2Combining rational expressions that have denominators with no common factors

Add and/or subtract each set of expressions. Make sure that you simplify each result.

  1. \(\frac{5}{x-2}-\frac{1}{x+7}\)
  2. \(\frac{2}{x-2}+\frac{x+3}{x+4}\)
  3. \(-\frac{1}{x}+\frac{x+1}{x^2-3x-10}\)
  4. \(\frac{x}{x^2+7x+12}-\frac{x}{x-3}\)

Solution
  1. \(\begin{aligned}[t] \amp\frac{5}{x-2}-\frac{1}{x+7}\\ \amp \phantom{={}} \phantom{={}} =\frac{5}{x-2} \cdot \frac{x+7}{x+7}-\frac{1}{x+7} \cdot \frac{x-2}{x-2}\\ \amp \phantom{={}} \phantom{={}} =\frac{5(x+7)-1 \cdot (x-2)}{(x-2)(x+7)}\\ \amp \phantom{={}} \phantom{={}} =\frac{5x+35-x+2}{(x-2)(x+7)}\\ \amp \phantom{={}} \phantom{={}} =\frac{4x+37}{(x-2)(x+7)} \end{aligned}\) \begin{equation*} \end{equation*}
  2. \(\begin{aligned}[t] \amp\frac{2}{x-2}+\frac{x+3}{x+4}\\ \amp \phantom{={}} \phantom{={}} =\frac{2}{x-2} \cdot \frac{x+4}{x+4}+\frac{x+3}{x+4} \cdot \frac{x-2}{x-2}\\ \amp \phantom{={}} \phantom{={}} =\frac{2(x+4)+(x+3)(x-2)}{(x-2)(x+4)}\\ \amp \phantom{={}} \phantom{={}} =\frac{2x+8+x^2-2x+3x-6}{(x-2)(x+4)}\\ \amp \phantom{={}} \phantom{={}} =\frac{x^2+3x+2}{(x-2)(x+4)}\\ \amp \phantom{={}} \phantom{={}} =\frac{(x+2)(x+1)}{(x-2)(x+4)} \end{aligned}\) \begin{equation*} \end{equation*}
  3. \(\begin{aligned}[t] \amp-\frac{1}{x}+\frac{x+1}{x^2-3x-10}\\ \amp \phantom{={}} \phantom{={}} =-\frac{1}{x}+\frac{x+1}{(x-5)(x+2)}\\ \amp \phantom{={}} \phantom{={}} =\frac{-1}{x} \cdot \frac{(x-5)(x+2)}{(x-5)(x+2)}+\frac{x+1}{(x-5)(x+2)} \cdot \frac{x}{x}\\ \amp \phantom{={}} \phantom{={}} =\frac{-1 \cdot (x-5)(x+2)+(x+1) \cdot x}{x(x-5)(x+2)}\\ \amp \phantom{={}} \phantom{={}} =\frac{-1 \cdot (x^2-3x-10)+x(x+1)}{x(x-5)(x+2)}\\ \amp \phantom{={}} \phantom{={}} =\frac{-x^2+3x+10+x^2+x}{x(x-5)(x+2)}\\ \amp \phantom{={}} \phantom{={}} =\frac{4x+10}{x(x-5)(x+2)}\\ \amp \phantom{={}} \phantom{={}} =\frac{2(x+5)}{x(x-5)(x+2)} \end{aligned}\) \begin{equation*} \end{equation*}
  4. \(\begin{aligned}[t] \amp\frac{x}{x^2+7x+12}-\frac{x}{x-3}\\ \amp \phantom{={}} \phantom{={}} =\frac{x}{(x+4)(x+3)}-\frac{x}{x-3}\\ \amp \phantom{={}} \phantom{={}} =\frac{x}{(x+4)(x+3)} \cdot \frac{x-3}{x-3}-\frac{x}{x-3} \cdot \frac{(x+4)(x+3)}{(x+4)(x+3)}\\ \amp \phantom{={}} \phantom{={}} =\frac{x(x-3)-x(x+4)(x+3)}{(x+4)(x+3)(x-3)}\\ \amp \phantom{={}} \phantom{={}} =\frac{x(x-3)-x(x^2+7x+12)}{(x+4)(x+3)(x-3)}\\ \amp \phantom{={}} \phantom{={}} =\frac{x^2-3x-x^3-7x^2-12x}{(x+4)(x+3)(x-3)}\\ \amp \phantom{={}} \phantom{={}} =\frac{-x^3-6x^2-15x}{(x+4)(x+3)(x-3)}\\ \amp \phantom{={}} \phantom{={}} =\frac{-x(x^2+6x+15)}{(x+4)(x+3)(x-3)} \end{aligned}\)
3Combining rational expressions that have denominators with both common factors and non-common factors

Add and/or subtract each set of expressions. Make sure that you simplify each result.

  1. \(\frac{x}{(2x-1)(x+5)}-\frac{x+1}{(2x-1)(x+16)}\)
  2. \(-\frac{1}{x^2+4x-12}+\frac{1}{x^2-7x-6}\)
  3. \(\frac{4}{x^2-3x+2}+\frac{4}{x-1}\)
  4. \(\frac{x-2}{x^2-5x}-\frac{2x}{x^2-10x+25}\)
  5. \(\frac{4}{x^3-4x^2+4x}-\frac{2}{x^2-2x}\)

Solution

Add and/or subtract each set of expressions. Make sure that you simplify each result.

  1. \(\begin{aligned}[t] \amp\frac{x}{(2x-1)(x+5)}-\frac{x+1}{2x-1(x+16)}\\ \amp \phantom{={}} \phantom{={}} =\frac{x}{(2x-1)(x+5)} \cdot \frac{x+16}{x+16} -\frac{x+1}{(2x-1)(x+16)} \cdot \frac{x+5}{x+5}\\ \amp \phantom{={}} \phantom{={}} =\frac{x(x+16)-(x+1)(x+5)}{(2x-1)(x+5)(x+16)}\\ \amp \phantom{={}} \phantom{={}} =\frac{x^2+16x-(x^2+6x+5)}{(2x-1)(x+5)(x+16)}\\ \amp \phantom{={}} \phantom{={}} =\frac{10x-5}{(2x-1)(x+5)(x+16)}\\ \amp \phantom{={}} \phantom{={}} =\frac{5(2x-1)}{(2x-1)(x+5)(x+16)}\\ \amp \phantom{={}} \phantom{={}} =\frac{2x-1}{2x-1} \cdot \frac{5}{(x+5)(x+16)}\\ \amp \phantom{={}} \phantom{={}} =\frac{5}{(x+5)(x+16)}, x \neq \frac{1}{2} \end{aligned}\) \begin{equation*} \end{equation*}
  2. \(\begin{aligned}[t] \amp\frac{1}{x^2-4x-12}+\frac{1}{x^2-7x-6}\\ \amp \phantom{={}} \phantom{={}} =\frac{1}{(x-6)(x+2)}+\frac{1}{(x-6)(x-1)}\\ \amp \phantom{={}} \phantom{={}} =\frac{1}{(x-6)(x+2)} \cdot \frac{x-1}{x-1} +\frac{1}{(x-6)(x-1)} \cdot \frac{x+2}{x+2}\\ \amp \phantom{={}} \phantom{={}} =\frac{1 \cdot (x-1)-1 \cdot (x+2)}{(x-6)(x+2)(x-1)}\\ \amp \phantom{={}} \phantom{={}} =\frac{-3}{(x-6)(x+2)(x-1)}\\ \amp \phantom{={}} \phantom{={}} =-\frac{3}{(x-6)(x+2)(x-1)} \end{aligned}\) \begin{equation*} \end{equation*}
  3. \(\begin{aligned}[t] \amp\frac{4}{x^2-3x+2}+\frac{4}{x-1}\\ \amp \phantom{={}} \phantom{={}} =\frac{4}{(x-2)(x-1)}+\frac{4}{x-1}\\ \amp \phantom{={}} \phantom{={}} =\frac{4}{(x-2)(x-1)}+\frac{4}{x-1} \cdot \frac{x-2}{x-2}\\ \amp \phantom{={}} \phantom{={}} =\frac{4+4(x-2)}{(x-2)(x-1)}\\ \amp \phantom{={}} \phantom{={}} =\frac{4+4x-8}{(x-2)(x-1)}\\ \amp \phantom{={}} \phantom{={}} =\frac{4x-4}{(x-2)(x-1)}\\ \amp \phantom{={}} \phantom{={}} =\frac{4(x-1)}{(x-2)(x-1)}\\ \amp \phantom{={}} \phantom{={}} =\frac{x-1}{x-1} \cdot \frac{4}{x-2}\\ \amp \phantom{={}} \phantom{={}} =\frac{4}{x-2}, x \neq 1 \end{aligned}\) \begin{equation*} \end{equation*}
  4. \(\begin{aligned}[t] \amp\frac{x-2}{x^2-5x}-\frac{2x}{x^2-10x+25}\\ \amp \phantom{={}} \phantom{={}} =\frac{x-2}{x(x-5)}-\frac{2x}{(x-5)(x-5)}\\ \amp \phantom{={}} \phantom{={}} =\frac{x-2}{x(x-5)} \cdot \frac{x-5}{x-5} -\frac{2x}{(x-5)(x-5)} \cdot \frac{x}{x}\\ \amp \phantom{={}} \phantom{={}} =\frac{(x-2)(x-5)-2x \cdot x}{x(x-5)(x-5)}\\ \amp \phantom{={}} \phantom{={}} =\frac{x^2-7x+10-2x^2}{x(x-5)(x-5)}\\ \amp \phantom{={}} \phantom{={}} =\frac{-x^2-7x+10}{x(x-5)^2} \end{aligned}\) \begin{equation*} \end{equation*}
  5. \(\begin{aligned}[t] \amp\frac{4}{x^3-4x^2+4x}-\frac{2}{x^2-2x}\\ \amp \phantom{={}} \phantom{={}} =\frac{4}{x(x^2-4x+4)}-\frac{2}{x(x-2)}\\ \amp \phantom{={}} \phantom{={}} =\frac{4}{x(x-2)(x-2)}+\frac{2}{x(x-2)}\\ \amp \phantom{={}} \phantom{={}} =\frac{4}{x(x-2)(x-2)}+\frac{2}{x(x-2)} \cdot \frac{x-2}{x-2}\\ \amp \phantom{={}} \phantom{={}} =\frac{4+2(x-2)}{x(x-2)(x-2)}\\ \amp \phantom{={}} \phantom{={}} =\frac{2x}{x(x-2)(x-2)}\\ \amp \phantom{={}} \phantom{={}} =\frac{x}{x} \cdot \frac{2}{(x-2)(x-2)}\\ \amp \phantom{={}} \phantom{={}} =\frac{2}{(x-2)^2}, x \neq 0 \end{aligned}\)
4Combining rational expressions that have denominators that include opposite factors

Add and/or subtract each set of expressions. Make sure that you simplify each result.

  1. \(\frac{3}{x-7}-\frac{6}{7-x}\)
  2. \(\frac{x}{x-9}+\frac{9}{9-x}\)
  3. \(\frac{28}{x^2-8x-33}+\frac{2}{11-x}\)
  4. \(\frac{x}{x^2-25}+\frac{2}{-x^2+x-20}\)

Solution

  1. \(\begin{aligned}[t] \frac{3}{x-7}-\frac{6}{7-x}\amp=\frac{3}{x-7}-\frac{6}{7-x} \cdot \frac{-1}{-1}\\ \amp=\frac{3}{x-7}-\frac{-6}{x-7}\\ \amp=\frac{3+6}{x-7}\\ \amp=\frac{9}{x-7} \end{aligned}\) \begin{equation*} \end{equation*}
  2. \(\begin{aligned}[t] \frac{x}{x-9}+\frac{9}{9-x}\amp=\frac{x}{x-9}+\frac{9}{9-x} \cdot \frac{-1}{-1}\\ \amp=\frac{x}{x-9}+\frac{-9}{x-9}\\ \amp=\frac{x-9}{x-9}\\ \amp=1, x \neq 9 \end{aligned}\) \begin{equation*} \end{equation*}
  3. \(\begin{aligned}[t] \amp\frac{28}{x^2-8x-33}+\frac{2}{11-x}\\ \amp \phantom{={}} \phantom{={}} =\frac{28}{(x-11)(x+3)}+\frac{2}{11-x}\\ \amp \phantom{={}} \phantom{={}} =\frac{28}{(x-11)(x+3)}+\frac{2}{11-x} \cdot \frac{-1}{-1}\\ \amp \phantom{={}} \phantom{={}} =\frac{28}{(x-11)(x+3)}+\frac{-2}{x-11}\\ \amp \phantom{={}} \phantom{={}} =\frac{28}{(x-11)(x+3)}+\frac{-2}{x-11} \cdot \frac{x+3}{x+3}\\ \amp \phantom{={}} \phantom{={}} =\frac{28-2(x+3)}{(x-11)(x+3)}\\ \amp \phantom{={}} \phantom{={}} =\frac{28-2x-6}{(x-11)(x+3)}\\ \amp \phantom{={}} \phantom{={}} =\frac{-2x+22}{(x-11)(x+3)}\\ \amp \phantom{={}} \phantom{={}} =\frac{-2(x-11)}{(x-11)(x+3)}\\ \amp \phantom{={}} \phantom{={}} =\frac{x-11}{x-11} \cdot \frac{-2}{x+3}\\ \amp \phantom{={}} \phantom{={}} =\frac{-2}{x+3}, x \neq 11 \end{aligned}\) \begin{equation*} \end{equation*}
  4. \(\begin{aligned}[t] \amp\frac{x}{x^2-25}+\frac{2}{-x^2+x-20}\\ \amp \phantom{={}} \phantom{={}} =\frac{x}{x^2-25}+\frac{2}{-x^2+x-20} \cdot \frac{-1}{-1}\\ \amp \phantom{={}} \phantom{={}} =\frac{x}{x^2-5}+\frac{-2}{x^2-x+20}\\ \amp \phantom{={}} \phantom{={}} =\frac{x}{(x+5)(x-5)}+\frac{-2}{(x+4)(x-5)}\\ \amp \phantom{={}} \phantom{={}} =\frac{x}{(x+5)(x-5)} \cdot \frac{x+4}{x+4}+\frac{-2}{(x+4)(x-5)} \cdot \frac{x+5}{x+5}\\ \amp \phantom{={}} \phantom{={}} =\frac{x(x+4)-2(x+5)}{(x+5)(x+4)(x-5)}\\ \amp \phantom{={}} \phantom{={}} =\frac{x^2+4x-2x-10}{(x+5)(x+4)(x-5)}\\ \amp \phantom{={}} \phantom{={}} =\frac{x^2+2x-10}{(x+5)(x+4)(x-5)} \end{aligned}\)