Chapter 8

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

Simplify each expression. \(\frac{9-t^{2}}{t^{2}+t-12}\)

NUMBER THEORY The ratio of 16 more than a number to 12 less than that number is 1 to 3 . What is the number?

Identify the type of function represented by each equation. Then graph the equation. \(y=|2 x|\)

State whether each equation represents a direct, joint, or inverse variation. Then name the constant of variation. \(v w=-18\)

Simplify each expression. $$ \frac{6}{a b}+\frac{8}{a} $$

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

Simplify each expression. \(\frac{3 x y z}{4 x z} \cdot \frac{6 x^{2}}{3 y^{2}}\)

NUMBER THEORY The sum of a number and 8 times its reciprocal is \(6 .\) Find the number(s).

Identify the type of function represented by each equation. Then graph the equation. \(y=2 x^{2}\)

State whether each equation represents a direct, joint, or inverse variation. Then name the constant of variation. \(y=-7 x\)

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

Simplify each expression. $$ \frac{5}{6 v}+\frac{7}{4 v} $$

Simplify each expression. \(\frac{-4 a b}{21 c} \cdot \frac{14 c^{2}}{18 a^{2}}\)

Solve each equation or inequality. Check your solutions. $$ \frac{b-4}{b-2}=\frac{b-2}{b+2}+\frac{1}{b-2} $$

Graph each rational function. $$ f(x)=\frac{1}{(x+3)^{2}} $$

Simplify each expression. $$ \frac{3 x}{4 y^{2}}-\frac{y}{6 x} $$

Simplify each expression. \(\frac{3}{5 d} \div\left(-\frac{9}{15 d f}\right)\)

Solve each equation or inequality. Check your solutions. $$ \frac{1}{n-2}=\frac{2 n+1}{n^{2}+2 n-8}+\frac{2}{n+4} $$

If \(y\) varies directly as \(x\) and \(y=9\) when \(x\) is \(-15,\) find \(y\) when \(x=21\)

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

Simplify each expression. $$ \frac{5}{a^{2} b}-\frac{7 a}{5 a^{2}} $$

Simplify each expression. \(\frac{p^{3}}{2 q} \div \frac{-p}{4 q}\)

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

If \(y\) varies directly as \(x\) and \(x=6\) when \(y=0.5,\) find \(y\) when \(x=10\)

Graph each rational function. $$ f(x)=\frac{x-1}{x-3} $$

Simplify each expression. $$ \frac{7}{y-8}-\frac{6}{8-y} $$

Simplify each expression. \(\frac{3 t^{2}}{t+2} \cdot \frac{t+2}{t^{2}}\)

Solve each equation or inequality. Check your solutions. $$ \frac{4}{z-2}-\frac{z+6}{z+1}=1 $$

Suppose \(y\) varies jointly as \(x\) and \(z .\) Find \(y\) when \(x=\frac{1}{2}\) and \(z=6,\) if \(y=45\) when \(x=6\) and \(z=10\).

Graph each rational function. $$ f(x)=\frac{x^{2}-36}{x+6} $$

Simplify each expression. $$ \frac{a}{a-4}-\frac{3}{4-a} $$

Simplify each expression. \(\frac{4 w+4}{3} \cdot \frac{1}{w+1}\)

Solve each equation or inequality. Check your solutions. $$ \frac{2}{3 y}+\frac{5}{6 y}>\frac{3}{4} $$

A woman painting a room will burn an average of 4.5 Calories per minute. Write an equation for the number of Calories burned in \(m\) minutes.

If \(y\) varies jointly as \(x\) and \(z\) and \(y=\frac{1}{8}\) when \(x=\frac{1}{2}\) and \(z=3,\) find \(y\) when \(x=6\) and \(z=\frac{1}{3}\).

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

Simplify each expression. $$ \frac{m}{m^{2}-4}+\frac{2}{3 m+6} $$

Simplify each expression. \(\frac{4 t^{2}-4}{9(t+1)^{2}} \cdot \frac{3 t+3}{2 t-2}\)

Solve each equation or inequality. Check your solutions. $$ \frac{1}{2 p}+\frac{3}{4 p}<\frac{1}{2} $$

PHYSICS For Exercises \(29-32,\) use the following information. Under certain conditions, when two objects collide, the objects are repellefrom each other with velocity given by the equation \(V_{f}=\frac{2 m_{1} v_{1}+v_{2}\left(m_{2}-m_{1}\right)}{m_{1}+m_{2}}\) In this equation \(m_{1}\) and \(m_{2}\) are the masses of the two objects, \(v_{1}\) and \(v_{2}\) are the initial speeds of the two objects, and \(V_{f}\) is the final speed of the second object. Let \(m_{2}\) be the independent variable, and let \(V_{f}\) be the dependent variable. Graph the function if \(m_{1}=5\) kilograms and \(v_{1}=15\) meters per second, and \(v_{2}=20\) meters per second.

If \(y\) varies inversely as \(x\) and \(y=2\) when \(x=25,\) find \(x\) when \(y=40\)

Simplify each expression. $$ \frac{y}{y+3}-\frac{6 y}{y^{2}-9} $$

Simplify each expression. \(\frac{3 p-21}{p^{2}-49} \cdot \frac{p^{2}-7 p}{3 p}\)

ACTINIIIES The band has 30 more members than the school chorale. If each group had 10 more members, the ratio of their membership would be \(3 : 2\) . How many members are in each group?

If \(y\) varies inversely as \(x\) and \(y=4\) when \(x=12,\) find \(y\) when \(x=5\)

Simplify each expression. $$ \frac{5}{x^{2}-3 x-28}+\frac{7}{2 x-14} $$

Simplify each expression. \(\frac{\frac{m^{3}}{3 n}}{-\frac{m^{4}}{9 n^{2}}}\)

PHYSICS For Exercises 31 and \(32,\) use the following information. The distance a spring stretches is related to the mass attached to the spring. This is represented by \(d=k m,\) where \(d\) is the distance, \(m\) is the mass, and \(k\) is the spring constant. When two springs with spring constants \(k_{1}\) and \(k_{2}\) are attached in a series, the resulting spring constant \(k\) is found by the equation \(\frac{1}{k}=\frac{1}{k_{1}}+\frac{1}{k_{2}}\) If one spring with constant of 12 centimeters per gram is attached in a series with another spring with constant of 8 centimeters per gram, find the resultant spring constant.

The shape of the Gateway Arch of the Jefferson National Expansion Memorial in St. Louis, Missouri, resembles the graph of the function \(f(x)=-0.00635 x^{2}+4.0005 x-0.07855,\) where \(x\) is in feet. Describe the shape of the Gateway Arch.