Chapter 7

In Exercises 11–18, graph the function. State the domain and range. $$ h(x)=\frac{6}{x-1} $$

Find the least common multiple of the expressions. \(x^2-25, x-5\)

Brass is an alloy composed of \(55 \%\) copper and \(45 \%\) zinc by weight. You have 25 ounces of copper. How many ounces of zinc do you need to make brass? (See Example 2.)

Find the product. $$ \frac{x^2(x-4)}{x-3} \cdot \frac{(x-3)(x+6)}{x^3} $$

In Exercises 11–18, graph the function. State the domain and range. $$ y=\frac{1}{x+2} $$

Find the least common multiple of the expressions. \(9 x^2-16,3 x^2+x-4\)

You have \(0.2\) liter of an acid solution whose acid concentration is 16 moles per liter. You want to dilute the solution with water so that its acid concentration is only 12 moles per liter. Use the given model to determine how many liters of water you should add to the solution.

Find the product. $$ \frac{x^3(x+5)}{x-9} \cdot \frac{(x-9)(x+8)}{3 x^3} $$

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

Find the least common multiple of the expressions. \(x^2+3 x-40, x-8\)

\(x=5, y=-4\)

Identify the least common denominator of the equation. $$\frac{x}{x+3}+\frac{1}{x}=\frac{3}{x}$$

Find the product. $$ \frac{x^2-3 x}{x-2} \cdot \frac{x^2+x-6}{x} $$

In Exercises 11–18, graph the function. State the domain and range. $$ f(x)=\frac{-2}{x-7} $$

Find the least common multiple of the expressions. \(x^2-2 x-63, x+7\)

Identify the least common denominator of the equation. $$\frac{5 x}{x-1}-\frac{7}{x}=\frac{9}{x}$$

Find the product. $$ \frac{x^2-4 x}{x-1} \cdot \frac{x^2+3 x-4}{2 x} $$

In Exercises 11–18, graph the function. State the domain and range. $$ g(x)=\frac{-3}{x-4}-1 $$

\(x=-3, y=8\)

Identify the least common denominator of the equation. $$\frac{2}{x+1}+\frac{x}{x+4}=\frac{1}{2}$$

Find the product. $$ \frac{x^2+3 x-4}{x^2+4 x+4} \cdot \frac{2 x^2+4 x}{x^2-4 x+3} $$

In Exercises 11–18, graph the function. State the domain and range. $$ y=\frac{10}{x+7}-5 $$

Identify the least common denominator of the equation. $$\frac{4}{x+9}+\frac{3 x}{2 x-1}=\frac{10}{3}$$

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

In Exercises 19-26, find the sum or difference. \(\frac{12}{5 x}-\frac{7}{6 x}\)

\(x=\frac{3}{4}, y=28\)

Solve the equation by using the LCD. Check your solution(s). $$\frac{3}{2}+\frac{1}{x}=2$$

Find the product. $$ \frac{x^2+5 x-36}{x^2-49} $$ \(\left(x^2-11 x+28\right)\)

Find the sum or difference. \(\frac{8}{3 x^2}+\frac{5}{4 x}\)

\(x=-4, y=-\frac{5}{4}\)

Solve the equation by using the LCD. Check your solution(s). $$\frac{2}{3 x}+\frac{1}{6}=\frac{4}{3 x}$$

Find the product. $$ \frac{x^2-x-12}{x^2-16} \cdot\left(x^2+2 x-8\right) $$

Find the sum or difference. \(\frac{3}{x+4}-\frac{1}{x+6}\)

\(x=-12, y=-\frac{1}{6}\)

Solve the equation by using the LCD. Check your solution(s). $$\frac{x-3}{x-4}+4=\frac{3 x}{x}$$

Find the sum or difference. \(\frac{9}{x-3}+\frac{2 x}{x+1}\)

\(x=\frac{5}{3}, y=-7\)

Solve the equation by using the LCD. Check your solution(s). $$\frac{2}{x-3}+\frac{1}{x}=\frac{x-1}{x-3}$$

Describe and correct the error in finding the product. $$ \begin{aligned} \frac{x^2-25}{3-x} \cdot \frac{x-3}{x+5} &=\frac{(x+5)(x-5)}{3-x} \cdot \frac{x-3}{x+5} \\ &=\frac{(x+5)(x-5)(x-5)}{(3-x)(x+5)} \\ &=x-5, x \neq 3, x \neq-5 \end{aligned} $$

Find the sum or difference. \(\frac{12}{x^2+5 x-24}+\frac{3}{x-3}\)

Solve the equation by using the LCD. Check your solution(s). $$\frac{6 x}{x+4}+4=\frac{2 x+2}{x-1}$$

Which rational expression is in simplifi ed form? (A) \(\frac{x^2-x-6}{x^2+3 x+2}\) (B) \(\frac{x^2+6 x+8}{x^2+2 x-3}\) (C) \(\frac{x^2-6 x+9}{x^2-2 x-3}\) (D) \(\frac{x^2+3 x-4}{x^2+x-2}\)

Find the sum or difference. \(\frac{x^2-5}{x^2+5 x-14}-\frac{x+3}{x+7}\)

Solve the equation by using the LCD. Check your solution(s). $$\frac{10}{x}+3=\frac{x+9}{x-4}$$

Find the product below by multiplying the numerators and denominators, then simplifying. Then fi nd the product by simplifying each expression, then multiplying. Which method do you prefer? Explain. $$ \frac{4 x^2 y}{2 x^3} \cdot \frac{12 y^4}{24 x^2} $$

In Exercises 25–32, graph the function. State the domain and range. $$ f(x)=\frac{x+4}{x-3} $$

Find the sum or difference. \(\frac{x+2}{x-4}+\frac{2}{x}+\frac{5 x}{3 x-1}\)

MODELING WITH MATHEMATICS The number \(y\) of songs that can be stored on an MP3 player varies inversely with the average size \(x\) of a song. A certain MP3 player can store 2500 songs when the average size of a song is 4 megabytes (MB). (See Example 4.) a. Make a table showing the number of songs that will fit on the MP3 player when the average size of a song is \(2 \mathrm{MB}, 2.5 \mathrm{MB}, 3 \mathrm{MB}\), and \(5 \mathrm{MB}\). b. What happens to the number of songs as the average song size increases?

Solve the equation by using the LCD. Check your solution(s). $$\frac{18}{x^2-3 x}-\frac{6}{x-3}=\frac{5}{x}$$

Compare the function $$ f(x)=\frac{(3 x-7)(x+6)}{(3 x-7)} \text { to the function } g(x)=x+6 $$