Chapter 11

Simplify the expression. If not possible, write already in simplest form. $$ \frac{t^{4}}{t^{2}(t+2)} $$

Does the equation model direct variation, inverse variation, or neither? $$ x y=9 $$

Add or subtract, then factor and simplify. $$ \frac{-12 y}{y^{2}-9 y+14}+\frac{84}{y^{2}-9 y+14} $$

Find the least common denominator of the pair of rational expressions. $$ \frac{4 x}{15}, \frac{3 x^{2}}{5} $$

Determine whether the equation follows from \(\frac{a}{b}=\frac{c}{d}\). $$ b a=d c $$

Find and correct the error. $$\frac{x+3}{x-3} \div \frac{4 x}{x^{2}-9}=\frac{x+3}{x-3} \cdot \frac{4 x}{(x+3)(x-3)}=\frac{4 x}{(x-3)^{2}}$$

Simplify the expression. If not possible, write already in simplest form. $$ \frac{8 n^{3}}{12 n^{4}+40 n^{2}} $$

Suppose \(y=6\) when \(x=4 .\) For the given type of variation, find an equation that relates \(x\) and \(y .\) $$\text{\(x\) and \(y\) vary directly.}$$

Add or subtract, then factor and simplify. $$ \frac{2 y+3}{y^{2}-4 y}-\frac{-y+15}{y^{2}-4 y} $$

Find the least common denominator of the pair of rational expressions. $$ \frac{17 y^{4}}{z^{2}}, \frac{8 z}{3 y} $$

Determine whether the equation follows from \(\frac{a}{b}=\frac{c}{d}\). $$ \frac{a}{d}=\frac{b}{c} $$

Write the product in simplest form. $$\frac{4 x}{3} \cdot \frac{1}{x}$$

Simplify the expression. If not possible, write already in simplest form. $$ \frac{18}{2 x+4} $$

Suppose \(y=6\) when \(x=4 .\) For the given type of variation, find an equation that relates \(x\) and \(y .\) $$\text{\(x\) and \(y\) vary inversely.}$$

Add or subtract, then factor and simplify. $$ \frac{10}{r^{2}+9 r+20}-\frac{-2 r}{r^{2}+9 r+20} $$

Find the least common denominator of the pair of rational expressions. $$ \frac{3}{c^{3}}, \frac{-5}{7 c^{5}} $$

Solve the equation by multiplying by the least common denominator. Check your solutions. \(\frac{1}{x}+\frac{x}{x+2}=1\)

Determine whether the equation follows from \(\frac{a}{b}=\frac{c}{d}\). $$ \frac{b}{a}=\frac{d}{c} $$

Write the product in simplest form. $$ \frac{9 x^{2}}{4} \cdot \frac{8}{18 x} $$

Simplify the expression. If not possible, write already in simplest form. $$ \frac{y^{7}-y^{3}}{y^{3}} $$

The variables \(x\) and \(y\) vary directly. Use the given values to write an equation that relates \(x\) and \(y .\) $$ x=3, y=9 $$

ADDING RATIONAL EXPRESSIONS. Simplify the expression. $$ \frac{7}{2 x}+\frac{x+2}{2 x} $$

Find the least common denominator of the pair of rational expressions. $$ \frac{10}{13 v^{7}}, \frac{10}{3 v^{5}} $$

Solve the equation by cross multiplying. Check your solutions. \(\frac{x}{5}=\frac{7}{3}\)

Solve the proportion using the reciprocal property. Check your solution. $$ \frac{3}{x}=\frac{1}{2} $$

Write the product in simplest form. $$\frac{7 d^{2}}{6 d} \cdot \frac{12 d^{2}}{2 d}$$

Simplify the expression. If not possible, write already in simplest form. $$ \frac{7-m}{m^{2}-49} $$

ADDING RATIONAL EXPRESSIONS. Simplify the expression. $$ \frac{2}{x+7}+\frac{5}{x+7} $$

The variables \(x\) and \(y\) vary directly. Use the given values to write an equation that relates \(x\) and \(y .\) $$ x=2, y=8 $$

Solve the equation by cross multiplying. Check your solutions. \(\frac{x}{10}=\frac{14}{5}\)

Find the least common denominator of the pair of rational expressions. $$ \frac{6 b}{5}, \frac{-5}{b} $$

Solve the proportion using the reciprocal property. Check your solution. $$ \frac{3}{4}=\frac{8}{3 c} $$

Write the product in simplest form. $$\frac{6 x}{14} \cdot \frac{2 x^{3}}{5 x^{5}}$$

Find the quotient. $$ \text { Divide }\left(3 y^{2}+22 y+7\right) \text { by }(y+7) $$

ADDING RATIONAL EXPRESSIONS. Simplify the expression. $$ \frac{4 t-1}{1-4 t}+\frac{2 t+3}{1-4 t} $$

The variables \(x\) and \(y\) vary directly. Use the given values to write an equation that relates \(x\) and \(y .\) $$ x=18, y=6 $$

Solve the equation by cross multiplying. Check your solutions. \(\frac{4}{x}=\frac{12}{5(x+2)}\)

Find the least common denominator of the pair of rational expressions. $$ \frac{x-1}{x-2}, \frac{x-3}{x-4} $$

Solve the proportion using the reciprocal property. Check your solution. $$ \frac{13}{z}=\frac{1}{3} $$

Write the product in simplest form. $$\frac{y}{16} \cdot \frac{4 y^{4}}{y^{2}}$$

Find the quotient. $$ \text { Divide }\left(x^{2}+5 x+6\right) \text { by }(x+3) $$

ADDING RATIONAL EXPRESSIONS. Simplify the expression. $$ \frac{4}{x+1}+\frac{2 x-2}{x+1} $$

The variables \(x\) and \(y\) vary directly. Use the given values to write an equation that relates \(x\) and \(y .\) $$ x=8, y=24 $$

Solve the equation by cross multiplying. Check your solutions. \(\frac{7}{x+1}=\frac{5}{x-3}\)

Find the least common denominator of the pair of rational expressions. $$ \frac{x+1}{15 x}, \frac{25}{18 x^{3}} $$

Solve the proportion using the cross product property. Check your solution. $$ \frac{5}{8}=\frac{c}{56} $$

Write the product in simplest form. $$\frac{-3}{x-4} \cdot \frac{x-4}{12(x-7)}$$

Find the quotient. $$ \text { Divide }\left(2 x^{2}-5 x-7\right) \text { by }(2 x-7) $$

ADDING RATIONAL EXPRESSIONS. Simplify the expression. $$ \frac{a+1}{15 a}+\frac{2 a-1}{15 a} $$

The variables \(x\) and \(y\) vary directly. Use the given values to write an equation that relates \(x\) and \(y .\) $$ x=36, y=12 $$