Chapter 8

Solve each equation. Check your solutions. $$ \frac{2}{d}+\frac{1}{4}=\frac{11}{12} $$

If \(y\) varies directly as \(x\) and \(y=18\) when \(x=15,\) find \(y\) when \(x=20\)

Determine the equations of any vertical asymptotes and the values of \(x\) for any holes in the graph of each rational function. $$ f(x)=\frac{3}{x^{2}-4 x+4} $$

Find the LCM of each set of polynomials. $$ 12 y^{2}, 6 x^{2} $$

Simplify each expression. \(\frac{45 m n^{3}}{20 n^{7}}\)

Solve each equation. Check your solutions. $$ t+\frac{12}{t}-8=0 $$

Suppose \(y\) varies jointly as \(x\) and \(z .\) Find \(y\) when \(x=9\) and \(z=-5\) if \(y=-90\) when \(z=15\) and \(x=-6 .\)

Determine the equations of any vertical asymptotes and the values of \(x\) for any holes in the graph of each rational function. $$ f(x)=\frac{x-1}{x^{2}+4 x-5} $$

Find the LCM of each set of polynomials. $$ 16 a b^{3}, 5 b^{2} a^{2}, 20 a c $$

Simplify each expression. \(\frac{a+b}{a^{2}-b^{2}}\)

Solve each equation. Check your solutions. $$ \frac{1}{x-1}+\frac{2}{x}=0 $$

If \(y\) varies inversely as \(x\) and \(y=-14\) when \(x=12,\) find \(x\) when \(y=21\)

Graph each rational function. $$ f(x)=\frac{x}{x+1} $$

Find the LCM of each set of polynomials. $$ x^{2}-2 x, x^{2}-4 $$

Simplify each expression. \(\frac{x^{2}+6 x+9}{x+3}\)

Solve each equation. Check your solutions. $$ \frac{12}{v^{2}-16}-\frac{24}{v-4}=3 $$

When a person swims underwater, the pressure in his or her ears varies directly with the depth at which he or she is swimming. Write a direct variation equation that represents this situation.

Graph each rational function. $$ f(x)=\frac{6}{(x-2)(x+3)} $$

Find the LCM of each set of polynomials. $$ x^{3}-4 x^{2}-5 x, x^{2}+6 x+5 $$

Simplify each expression. \(\frac{36 c^{3} d^{2}}{54 c d^{5}}\)

Solve each equation. Check your solutions. $$ \frac{w}{w-1}+w=\frac{4 w-3}{w-1} $$

Graph each rational function. $$ f(x)=\frac{4}{(x-1)^{2}} $$

Simplify each expression. $$ \frac{2}{x^{2} y}-\frac{x}{y} $$

Identify all values of \(y\) for which \(\frac{y-4}{y^{2}-4 y-12}\) is undefined. A. -2, 4, 6 B. -6, -4, 2 C. -2, 0, 6 D. -2, 6

Solve each equation. Check your solutions. $$ \frac{4 n^{2}}{n^{2}-9}-\frac{2 n}{n+3}=\frac{3}{n-3} $$

Graph each rational function. $$ f(x)=\frac{x-5}{x+1} $$

Simplify each expression. $$ \frac{7 a}{15 b^{2}}-\frac{b}{18 a b} $$

Simplify each expression. \(\frac{9 y^{2}-6 y^{3}}{2 y^{2}+5 y-12}\)

Work A worker can power wash a wall of a certain size in 5 hours. Another worker can do the same job in 4 hours. If the workers work together, how long would it take to do the job? Determine whether your answer is reasonable.

Simplify each expression. $$ \frac{5}{3 m}-\frac{2}{7 m}-\frac{1}{2 m} $$

Simplify each expression. \(\frac{b^{3}-a^{3}}{a^{2}-b^{2}}\)

Solve each inequality. $$ \frac{4}{c+2}>1 $$

If \(y\) varies directly as \(x\) and \(y=15\) when \(x=3,\) find \(y\) when \(x=12\).

Graph each rational function. $$ f(x)=\frac{x+2}{x^{2}-x-6} $$

Simplify each expression. $$ \frac{3 x}{5}-\frac{1}{2 x^{2}}+\frac{3}{4 x} $$

Simplify each expression. \(\frac{2 a^{2}}{5 b^{2} c} \cdot \frac{3 b c^{3}}{8 a^{2}}\)

Solve each inequality. $$ \frac{1}{3 v}+\frac{1}{4 v}<\frac{1}{2} $$

If \(y\) varies directly as \(x\) and \(y=8\) when \(x=6,\) find \(y\) when \(x=15\)

ELECTRICITY For Exercises \(9-12\) , use the following information. The current \(I\) in amperes in an electrical circuit with three resistors in series is given by the equation \(I=\frac{V}{R_{1}+R_{2}+R_{3}},\) where \(V\) is the voltage in volts in the circuit and \(R_{1}, R_{2},\) and \(R_{3}\) are the resistances in ohms of the three resistors. Let \(R_{1}\) be the independent variable, and let I be the dependent variable. Graph the function if \(V=120\) volts, \(R_{2}=25\) ohms, and \(R_{3}=75\) ohms.

Simplify each expression. $$ \frac{6}{d^{2}+4 d+4}+\frac{5}{d+2} $$

Simplify each expression. \(\frac{3 t+6}{7 t-7} \cdot \frac{14 t-14}{5 t+10}\)

Solve each equation or inequality. Check your solutions. $$ \frac{y}{y+1}=\frac{2}{3} $$

Suppose \(y\) varies jointly as \(x\) and \(z .\) Find \(y\) when \(x=2\) and \(z=27\) if \(y=192\) when \(x=8\) and \(z=6 .\)

Simplify each expression. $$ \frac{a}{a^{2}-a-20}+\frac{2}{a+4} $$

Simplify each expression. \(\frac{35}{16 x^{2}} \div \frac{21}{4 x}\)

Solve each equation or inequality. Check your solutions. $$ \frac{p}{p-2}=\frac{2}{5} $$

If \(y\) varies jointly as \(x\) and \(z\) and \(y=80\) when \(x=5\) and \(z=8,\) find \(y\) when \(x=16\) and \(z=2\)

Simplify each expression. $$ \frac{1}{x^{2}-4}+\frac{x}{x+2} $$

Simplify each expression. \(\frac{20 x y^{3}}{21} \div \frac{15 x^{3} y^{2}}{14}\)

Solve each equation or inequality. Check your solutions. $$ s+5=\frac{6}{s} $$