Chapter 7

A fraction that contains a fraction in its numerator or denominator is called a(n) ______.

VOCABULARY Explain how direct variation equations and inverse variation equations are different.

When can you solve a rational equation by cross multiplying? Explain.

Describe how to multiply and divide two rational expressions

Is \(f(x)=\frac{-3 x+5}{2^x+1}\) a rational function? Explain your reasoning.

Explain how adding and subtracting rational expressions is similar to adding and subtracting numerical fractions.

A student solves the equation \(\frac{4}{x-3}=\frac{x}{x-3}\) and obtains the solutions 3 and 4 . Are either of these extraneous solutions? Explain.

In Exercises 3-10, graph the function. Compare the graph with the graph of \(f(x)=\frac{1}{x}\). $$ g(x)=\frac{3}{x} $$

In Exercises 3-8, find the sum or difference. \(\frac{15}{4 x}+\frac{5}{4 x}\)

\(y=\frac{2}{x}\)

Solve the equation by cross multiplying. Check your solution(s). $$\frac{4}{2 x}=\frac{5}{x+6}$$

Simplify the expression, if possible. $$ \frac{2 x^2}{3 x^2-4 x} $$

In Exercises 3-10, graph the function. Compare the graph with the graph of \(f(x)=\frac{1}{x}\). $$ g(x)=\frac{10}{x} $$

Find the sum or difference. \(\frac{x}{16 x^2}-\frac{4}{16 x^2}\)

Solve the equation by cross multiplying. Check your solution(s). $$\frac{9}{3 x}=\frac{4}{x+2}$$

Simplify the expression, if possible. $$ \frac{7 x^3-x^2}{2 x^3} $$

In Exercises 3-10, graph the function. Compare the graph with the graph of \(f(x)=\frac{1}{x}\). $$ g(x)=\frac{-5}{x} $$

Find the sum or difference. \(\frac{9}{x+1}-\frac{2 x}{x+1}\)

\(\frac{y}{x}=8\)

Solve the equation by cross multiplying. Check your solution(s). $$\frac{6}{x-1}=\frac{9}{x+1}$$

Simplify the expression, if possible. $$ \frac{x^2-3 x-18}{x^2-7 x+6} $$

In Exercises 3-10, graph the function. Compare the graph with the graph of \(f(x)=\frac{1}{x}\). $$ g(x)=\frac{-9}{x} $$

Find the sum or difference. \(\frac{3 x^2}{x-8}+\frac{6 x}{x-8}\)

Solve the equation by cross multiplying. Check your solution(s). $$\frac{8}{3 x-2}=\frac{2}{x-1}$$

Simplify the expression, if possible. $$ \frac{x^2+13 x+36}{x^2-7 x+10} $$

In Exercises 3-10, graph the function. Compare the graph with the graph of \(f(x)=\frac{1}{x}\). $$ g(x)=\frac{15}{x} $$

Find the sum or difference. \(\frac{5 x}{x+3}+\frac{15}{x+3}\)

Solve the equation by cross multiplying. Check your solution(s). $$\frac{x}{2 x+7}=\frac{x-5}{x-1}$$

Simplify the expression, if possible. $$ \frac{x^2+11 x+18}{x^3+8} $$

In Exercises 3-10, graph the function. Compare the graph with the graph of \(f(x)=\frac{1}{x}\). $$ g(x)=\frac{-12}{x} $$

Find the sum or difference. \(\frac{4 x^2}{2 x-1}-\frac{1}{2 x-1}\)

Solve the equation by cross multiplying. Check your solution(s). $$\frac{-2}{x-1}=\frac{x-8}{x+1}$$

Simplify the expression, if possible. $$ \frac{x^2-7 x+12}{x^3-27} $$

In Exercises 3-10, graph the function. Compare the graph with the graph of \(f(x)=\frac{1}{x}\). $$ g(x)=\frac{-0.5}{x} $$

In Exercises 9-16, find the least common multiple of the expressions. \(3 x, 3(x-2)\)

Solve the equation by cross multiplying. Check your solution(s). $$\frac{x^2-3}{x+2}=\frac{x-3}{2}$$

Simplify the expression, if possible. $$ \frac{32 x^4-50}{4 x^3-12 x^2-5 x+15} $$

In Exercises 3-10, graph the function. Compare the graph with the graph of \(f(x)=\frac{1}{x}\). $$ g(x)=\frac{0.1}{x} $$

Find the least common multiple of the expressions. \(2 x^2, 4 x+12\)

\(x y=\frac{1}{5}\)

Solve the equation by cross multiplying. Check your solution(s). $$\frac{-1}{x-3}=\frac{x-4}{x^2-27}$$

Simplify the expression, if possible. $$ \frac{3 x^3-3 x^2+7 x-7}{27 x^4-147} $$

In Exercises 11–18, graph the function. State the domain and range. $$ g(x)=\frac{4}{x}+3 $$

Find the least common multiple of the expressions. \(2 x, 2 x(x-5)\)

So far in your volleyball practice, you have put into play 37 of the 44 serves you have attempted. Solve the equation \(\frac{90}{100}=\frac{37+x}{44+x}\) to find the number of consecutive serves you need to put into play in order to raise your serve percentage to \(90 \%\).

Find the product. $$ \frac{4 x y^3}{x^2 y} \cdot \frac{y}{8 x} $$

In Exercises 11–18, graph the function. State the domain and range. $$ y=\frac{2}{x}-3 $$

Find the least common multiple of the expressions. \(24 x^2, 8 x^2-16 x\)

So far this baseball season, you have 12 hits out of 60 times at-bat. Solve the equation \(0.360=\frac{12+x}{60+x}\) to find the number of consecutive hits you need to raise your batting average to \(0.360\).

Find the product. $$ \frac{48 x^5 y^3}{y^4} \cdot \frac{x^2 y}{6 x^3 y^2} $$