8.4 Add and Subtract Rational Expressions with Unlike Denominators

If you miss a problem, go back to the section listed and review the material.

Add: $\frac{7}{10} + \frac{8}{15} .$
If you missed this problem, review the reference.

Subtract: $6 \left(2 x + 1\right) - 4 \left(x - 5\right) .$
If you missed this problem, review the reference.

Find the Greatest Common Factor of $9 x^{2} y^{3}$ and $12 x y^{5}$.
If you missed this problem, review the reference.

Factor completely $−48 n - 12$.
If you missed this problem, review the reference.

Find the LCD for $\frac{8}{x^{2} - 2 x - 3} , \frac{3 x}{x^{2} + 4 x + 3}$.

Find the LCD for $\frac{2}{x^{2} - x - 12} , \frac{1}{x^{2} - 16}$.

Find the LCD for $\frac{x}{x^{2} + 8 x + 15} , \frac{5}{x^{2} + 9 x + 18}$.

Rewrite as equivalent rational expressions with denominator $\left(x + 3\right) \left(x - 4\right) \left(x + 4\right)$:
$\frac{2}{x^{2} - x - 12} , \frac{1}{x^{2} - 16} .$

Rewrite as equivalent rational expressions with denominator $\left(x + 3\right) \left(x + 5\right) \left(x + 6\right)$:
$\frac{x}{x^{2} + 8 x + 15} , \frac{5}{x^{2} + 9 x + 18} .$

Add: $\frac{11}{30} + \frac{7}{12} .$

Add: $\frac{3}{8} + \frac{9}{20} .$

Add: $\frac{2}{15 a^{2} b} + \frac{5}{6 a b^{2}} .$

Add: $\frac{5}{16 c} + \frac{3}{8 c d^{2}} .$

Add: $\frac{2}{x - 2} + \frac{5}{x + 3} .$

Add: $\frac{4}{m + 3} + \frac{3}{m + 4} .$

Add: $\frac{5 x}{x y - y^{2}} + \frac{2 x}{x^{2} - y^{2}} .$

Add: $\frac{7}{2 m + 6} + \frac{4}{m^{2} + 4 m + 3} .$

Add: $\frac{1}{m^{2} - m - 2} + \frac{5 m}{m^{2} + 3 m + 2} .$

Add: $\frac{2 n}{n^{2} - 3 n - 10} + \frac{6}{n^{2} + 5 n + 6} .$

Subtract: $\frac{y}{y + 4} - \frac{y - 2}{y - 5} .$

Subtract: $\frac{z + 3}{z + 2} - \frac{z}{z + 3} .$

Subtract: $\frac{2 x}{x^{2} - 4} - \frac{1}{x + 2} .$

Subtract: $\frac{3}{z + 3} - \frac{6 z}{z^{2} - 9} .$

Subtract: $\frac{3 x - 1}{x^{2} - 5 x - 6} - \frac{2}{6 - x} .$

Subtract: $\frac{−2 y - 2}{y^{2} + 2 y - 8} - \frac{y - 1}{2 - y} .$

Subtract: $\frac{2 x + 1}{x - 7} - 3 .$

Subtract: $\frac{4 y + 3}{2 y - 1} - 5 .$

Simplify: $\frac{v}{v + 1} + \frac{3}{v - 1} - \frac{6}{v^{2} - 1} .$

Simplify: $\frac{3 w}{w + 2} + \frac{2}{w + 7} - \frac{17 w + 4}{w^{2} + 9 w + 14} .$

Find the LCD.

$\frac{5}{x^{2} - 2 x - 8} , \frac{2 x}{x^{2} - x - 12}$

Find the LCD.

$\frac{9}{z^{2} + 2 z - 8} , \frac{4 z}{z^{2} - 4}$

Find the LCD.

$\frac{4}{b^{2} + 6 b + 9} , \frac{2 b}{b^{2} - 2 b - 15}$

Find the LCD.

$\frac{2}{3 d^{2} + 14 d - 5} , \frac{5 d}{3 d^{2} - 19 d + 6}$

Write as equivalent rational expressions with the given LCD.

$\frac{5}{x^{2} - 2 x - 8} , \frac{2 x}{x^{2} - x - 12}$
LCD $\left(x - 4\right) \left(x + 2\right) \left(x + 3\right)$

Write as equivalent rational expressions with the given LCD.

$\frac{9}{z^{2} + 2 z - 8} , \frac{4 z}{z^{2} - 4}$
LCD $\left(z - 2\right) \left(z + 4\right) \left(z + 2\right)$

Write as equivalent rational expressions with the given LCD.

$\frac{4}{b^{2} + 6 b + 9} , \frac{2 b}{b^{2} - 2 b - 15}$
LCD $\left(b + 3\right) \left(b + 3\right) \left(b - 5\right)$

Write as equivalent rational expressions with the given LCD.

$\frac{2}{3 d^{2} + 14 d - 5} , \frac{5 d}{3 d^{2} - 19 d + 6}$
LCD $\left(3 d - 1\right) \left(d + 5\right) \left(d - 6\right)$

Add.

$\frac{5}{24} + \frac{11}{36}$

Add.

$\frac{9}{20} + \frac{11}{30}$

Add.

$\frac{7}{10 x^{2} y} + \frac{4}{15 x y^{2}}$

Add.

$\frac{1}{2 m} + \frac{7}{8 m^{2} n}$

Add.

$\frac{3}{r + 4} + \frac{2}{r - 5}$

Add.

$\frac{8}{t + 5} + \frac{6}{t - 5}$

Add.

$\frac{5}{3 w - 2} + \frac{2}{w + 1}$

Add.

$\frac{2 y}{y + 3} + \frac{3}{y - 1}$

Add.

$\frac{5 b}{a^{2} b - 2 a^{2}} + \frac{2 b}{b^{2} - 4}$

Add.

$\frac{2 m}{3 m - 3} + \frac{5 m}{m^{2} + 3 m - 4}$

Add.

$\frac{3}{n^{2} + 3 n - 18} + \frac{4 n}{n^{2} + 8 n + 12}$

Add.

$\frac{3 r}{r^{2} + 7 r + 6} + \frac{9}{r^{2} + 4 r + 3}$

Subtract.

$\frac{t}{t - 6} - \frac{t - 2}{t + 6}$

Subtract.

$\frac{w + 2}{w + 4} - \frac{w}{w - 2}$

Subtract.

$\frac{y - 4}{y + 1} - \frac{1}{y + 7}$

Subtract.

$\frac{5 a}{a + 3} - \frac{a + 2}{a + 6}$

Subtract.

$\frac{6 c}{c^{2} - 25} - \frac{3}{c + 5}$

Subtract.

$\frac{6}{m + 6} - \frac{12 m}{m^{2} - 36}$

Subtract.

$\frac{−9 p - 17}{p^{2} - 4 p - 21} - \frac{p + 1}{7 - p}$

Subtract.

$\frac{−2 r - 16}{r^{2} + 6 r - 16} - \frac{5}{2 - r}$

Subtract.

$\frac{5 v - 2}{v + 3} - 4$

Subtract.

$\frac{2 x + 7}{10 x - 1} + 3$

Add and subtract.

$\frac{5 a}{a - 2} + \frac{9}{a} - \frac{2 a + 18}{a^{2} - 2 a}$

Add and subtract.

$\frac{c}{c + 2} + \frac{5}{c - 2} - \frac{10 c}{c^{2} - 4}$

Simplify.

$\frac{6 a}{3 a b + b^{2}} + \frac{3 a}{9 a^{2} - b^{2}}$

Simplify.

$\frac{6 d}{d^{2} - 64} - \frac{3}{d - 8}$

Simplify.

$\frac{4 m}{m^{2} + 6 m - 7} + \frac{2}{m^{2} + 10 m + 21}$

Simplify.

$\frac{−5 n - 5}{n^{2} + n - 6} + \frac{n + 1}{2 - n}$

Simplify.

$\frac{7}{15 p} + \frac{5}{18 p q}$

Simplify.

$\frac{4}{x - 2} + \frac{3}{x + 5}$

Simplify.

$\frac{2 q + 7}{q + 4} - 2$

Simplify.

$\frac{z + 2}{z - 5} - \frac{z}{z + 1}$

Simplify.

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

Decorating cupcakes Victoria can decorate an order of cupcakes for a wedding in $t$ hours, so in 1 hour she can decorate $\frac{1}{t}$ of the cupcakes. It would take her sister 3 hours longer to decorate the same order of cupcakes, so in 1 hour she can decorate $\frac{1}{t + 3}$ of the cupcakes.

1. ⓐ Find the fraction of the decorating job that Victoria and her sister, working together, would complete in one hour by adding the r…

Felipe thinks $\frac{1}{x} + \frac{1}{y}$ is $\frac{2}{x + y} .$

1. ⓐ Choose numerical values for x and y and evaluate $\frac{1}{x} + \frac{1}{y} .$
2. ⓑ Evaluate $\frac{2}{x + y}$ for the same values of x and y you used in part (a).
3. ⓒ Explain why Felipe is wrong.
4. ⓓ Find the correct expression for $\frac{1}{x} + \frac{1}{y} .$