7.4 Solve Rational Equations

Before you get started, take this readiness quiz.

Solve: $\frac{1}{6} x + \frac{1}{2} = \frac{1}{3} .$
If you missed this problem, review the reference.

Solve: $n^{2} - 5 n - 36 = 0 .$
If you missed this problem, review the reference.

Solve the formula $5 x + 2 y = 10$ for $y .$
If you missed this problem, review the reference.

Solve: $\frac{1}{y} + \frac{2}{3} = \frac{1}{5} .$

Solve: $\frac{2}{3} + \frac{1}{5} = \frac{1}{x} .$

Solve: $1 - \frac{2}{x} = \frac{15}{x^{2}} .$

Solve: $1 - \frac{4}{y} = \frac{12}{y^{2}} .$

Solve: $\frac{2}{x + 1} + \frac{1}{x - 1} = \frac{1}{x^{2} - 1} .$

Solve: $\frac{5}{y + 3} + \frac{2}{y - 3} = \frac{5}{y^{2} - 9} .$

Solve: $\frac{x + 13}{x^{2} - 7 x + 10} = \frac{6}{x - 5} - \frac{4}{x - 2} .$

Solve: $\frac{y - 6}{y^{2} + 3 y - 4} = \frac{2}{y + 4} + \frac{7}{y - 1} .$

Solve: $\frac{x}{x + 4} = \frac{32}{x^{2} - 16} + 5 .$

Solve: $\frac{y}{y + 8} = \frac{128}{y^{2} - 64} + 9 .$

Solve: $\frac{y}{5 y - 10} - \frac{5}{3 y + 6} = \frac{2 y^{2} - 19 y + 54}{15 y^{2} - 60} .$

Solve: $\frac{z}{2 z + 8} - \frac{3}{4 z - 8} = \frac{3 z^{2} - 16 z - 16}{8 z^{2} + 16 z - 64} .$

Solve: $\frac{15}{x^{2} + x - 6} - \frac{3}{x - 2} = \frac{2}{x + 3} .$

Solve: $\frac{5}{x^{2} + 2 x - 3} - \frac{3}{x^{2} + x - 2} = \frac{1}{x^{2} + 5 x + 6} .$

For rational function, $f \left(x\right) = \frac{8 - x}{x^{2} - 7 x + 12} ,$ ⓐ find the domain of the function ⓑ solve $f \left(x\right) = 3$ ⓒ find the points on the graph at this function value.

For rational function, $f \left(x\right) = \frac{x - 1}{x^{2} - 6 x + 5} ,$ ⓐ find the domain of the function ⓑ solve $f \left(x\right) = 4$ ⓒ find the points on the graph at this function value.

Solve: $m = \frac{y - 5}{x - 4}$for $y .$

Solve: $m = \frac{y - 1}{x + 5}$ for $y .$

Solve: $\frac{1}{a} + \frac{1}{b} = c$ for a.

Solve: $\frac{2}{x} + \frac{1}{3} = \frac{1}{y}$ for y.

Solve each rational equation.

$\frac{1}{a} + \frac{2}{5} = \frac{1}{2}$

Solve each rational equation.

$\frac{4}{5} + \frac{1}{4} = \frac{2}{v}$

Solve each rational equation.

$1 - \frac{2}{m} = \frac{8}{m^{2}}$

Solve each rational equation.

$1 + \frac{9}{p} = \frac{−20}{p^{2}}$

Solve each rational equation.

$\frac{5}{3 v - 2} = \frac{7}{4 v}$

Solve each rational equation.

$\frac{3}{x + 4} + \frac{7}{x - 4} = \frac{8}{x^{2} - 16}$

Solve each rational equation.

$\frac{8}{z - 10} - \frac{7}{z + 10} = \frac{5}{z^{2} - 100}$

Solve each rational equation.

$\frac{−10}{q - 2} - \frac{7}{q + 4} = 1$

Solve each rational equation.

$\frac{v - 10}{v^{2} - 5 v + 4} = \frac{3}{v - 1} - \frac{6}{v - 4}$

Solve each rational equation.

$\frac{x - 10}{x^{2} + 8 x + 12} = \frac{3}{x + 2} + \frac{4}{x + 6}$

Solve each rational equation.

$\frac{b + 3}{3 b} + \frac{b}{24} = \frac{1}{b}$

Solve each rational equation.

$\frac{d}{d + 3} = \frac{18}{d^{2} - 9} + 4$

Solve each rational equation.

$\frac{n}{n + 2} - 3 = \frac{8}{n^{2} - 4}$

Solve each rational equation.

$\frac{q}{3 q - 9} - \frac{3}{4 q + 12} = \frac{7 q^{2} + 6 q + 63}{24 q^{2} - 216}$

Solve each rational equation.

$\frac{s}{2 s + 6} - \frac{2}{5 s + 5} = \frac{5 s^{2} - 3 s - 7}{10 s^{2} + 40 s + 30}$

Solve each rational equation.

$\frac{2}{x^{2} + 2 x - 8} - \frac{1}{x^{2} + 9 x + 20} = \frac{4}{x^{2} + 3 x - 10}$

Solve each rational equation.

$\frac{3}{x^{2} - 5 x - 6} + \frac{3}{x^{2} - 7 x + 6} = \frac{6}{x^{2} - 1}$

Solve Rational Equations that Involve Functions

For rational function, $f \left(x\right) = \frac{x - 2}{x^{2} + 6 x + 8} ,$ ⓐ find the domain of the function ⓑ solve $f \left(x\right) = 5$ ⓒ find the points on the graph at this function value.

Solve Rational Equations that Involve Functions

For rational function, $f \left(x\right) = \frac{2 - x}{x^{2} - 7 x + 10} ,$ ⓐ find the domain of the function ⓑ solve $f \left(x\right) = 2$ ⓒ find the points on the graph at this function value.

Solve.

$\frac{C}{r} = 2 \pi$ for $r .$

Solve.

$\frac{v + 3}{w - 1} = \frac{1}{2}$ for $w .$

Solve.

$a = \frac{b + 3}{c - 2}$ for $c .$

Solve.

$\frac{1}{p} + \frac{2}{q} = 4$ for $p .$

Solve.

$\frac{2}{v} + \frac{1}{5} = \frac{3}{w}$ for $w .$

Solve.

$\frac{m + 3}{n - 2} = \frac{4}{5}$ for $n .$

Solve.

$\frac{E}{c} = m^{2}$ for $c .$

Solve.

$\frac{3}{x} - \frac{5}{y} = \frac{1}{4}$ for $y .$

Your class mate is having trouble in this section. Write down the steps you would use to explain how to solve a rational equation.