Chapter 7

Solve or simplify, whichever is appropriate. $$\frac{x^{2}+4 x-2}{x^{2}-2 x-8}-1-\frac{4}{x-4}$$

denominators are opposites, or additive inverses. Add or subtract as indicated. Simplify the result, if possible. $$\frac{10}{x+3}-\frac{2}{-x-3}$$

It normally takes 2 hours to fill a swimming pool. The pool has developed a slow leak. If the pool were full, it would take 10 hours for all the water to leak out. If the pool is empty, how long will it take to fill it?

Simplify each rational expression. If the rational expression cannot be simplified, so state. $$\frac{x^{3}-8}{x-2}$$

Add or subtract as indicated. Simplify the result, if possible. $$\frac{3}{x^{2}-1}+\frac{4}{(x+1)^{2}}$$

Divide as indicated. $$\frac{x^{2}-25}{2 x-2}+\frac{x^{2}+10 x+25}{x^{2}+4 x-5}$

Solve or simplify, whichever is appropriate. $$5 y^{-2}+1=6 y^{-1}$$

denominators are opposites, or additive inverses. Add or subtract as indicated. Simplify the result, if possible. $$\frac{11}{x+7}-\frac{5}{-x-7}$$

Two investments have interest rates that differ by \(1 \% .\) An investment for 1 year at the lower rate earns 175 . The same principal invested for a year at the higher rate earns 200 dollars . What are the two interest rates?

Simplify each rational expression. If the rational expression cannot be simplified, so state. $$\frac{x^{3}-125}{x^{2}-25}$$

Add or subtract as indicated. Simplify the result, if possible. $$\frac{6}{x^{2}-4}+\frac{2}{(x+2)^{2}}$$

Divide as indicated. $$\frac{x^{2}-4}{x^{2}+3 x-10} \div \frac{x^{2}+5 x+6}{x^{2}+8 x+15}$$

Solve or simplify, whichever is appropriate. $$3 y^{-2}+1=4 y^{-1}$$

denominators are opposites, or additive inverses. Add or subtract as indicated. Simplify the result, if possible. $$\frac{y}{y-1}-\frac{1}{1-y}$$

Factor: \(25 x^{2}-81 .\) (Section 6.4, Example 1)

Simplify each rational expression. If the rational expression cannot be simplified, so state. $$\frac{(x-4)^{2}}{x^{2}-16}$$

What is a complex rational expression? Give an example with your explanation.

Add or subtract as indicated. Simplify the result, if possible. $$\frac{3 x}{x^{2}+3 x-10}-\frac{2 x}{x^{2}+x-6}$$

Divide as indicated. $$\frac{y^{3}+y}{y^{2}-y} \div \frac{y^{3}-y^{2}}{y^{2}-2 y+1}$$

Solve or simplify, whichever is appropriate. $$\frac{3}{y+1}-\frac{1}{1-y}=\frac{10}{y^{2}-1}$$

denominators are opposites, or additive inverses. Add or subtract as indicated. Simplify the result, if possible. $$\frac{y}{y-4}-\frac{4}{4-y}$$

Solve: \(x^{2}-12 x+36=0 .\) (Section 6.6, Example 4)

Simplify each rational expression. If the rational expression cannot be simplified, so state. $$\frac{(x+5)^{2}}{x^{2}-25}$$

Describe two ways to simplify \(\frac{\frac{3}{x}+\frac{2}{x^{2}}}{\frac{1}{x^{2}}+\frac{2}{x}}\).

Add or subtract as indicated. Simplify the result, if possible. $$\frac{x}{x^{2}-2 x-24}-\frac{x}{x^{2}-7 x+6}$$

Divide as indicated. $$\frac{3 y^{2}-12}{y^{2}+4 y+4} \div \frac{y^{3}-2 y^{2}}{y^{2}+2 y}$$

Solve or simplify, whichever is appropriate. $$\frac{4}{y-2}-\frac{1}{2-y}=\frac{25}{y+6}$$

denominators are opposites, or additive inverses. Add or subtract as indicated. Simplify the result, if possible. $$\frac{3-x}{x-7}-\frac{2 x-5}{7-x}$$

Graph: \(y=-\frac{2}{3} x+4 .\) (Section 3.4, Example 3)

Simplify each rational expression. If the rational expression cannot be simplified, so state. $$\frac{x}{x+1}$$

Which method do you prefer for simplifying complex rational expressions? Why?

Add or subtract as indicated. Simplify the result, if possible. $$\frac{y}{y^{2}+2 y+1}+\frac{4}{y^{2}+5 y+4}$$

Divide as indicated. $$\frac{y^{2}+5 y+4}{y^{2}+12 y+32}+\frac{y^{2}-12 y+35}{y^{2}+3 y-40}$$

denominators are opposites, or additive inverses. Add or subtract as indicated. Simplify the result, if possible. $$\frac{4-x}{x-9}-\frac{3 x-8}{9-x}$$

Will help you prepare for the material covered in the next section. a. If \(y=k x\) find the value of \(k\) using \(x=2\) and \(y=64\) b. Substitute the value for \(k\) into \(y=k x\) and write the resulting equation. c. Use the equation from part (b) to find \(y\) when \(x=5\)

Simplify each rational expression. If the rational expression cannot be simplified, so state. $$\frac{x}{x+7}$$

Determine whether statement "makes sense" or "does not make sense" and explain your reasoning. I simplified \(\frac{\frac{1}{2}+\frac{x}{3}}{4}\) by multiplying the numerator by 6.

Add or subtract as indicated. Simplify the result, if possible. $$\frac{y}{y^{2}+5 y+6}+\frac{4}{y^{2}-y-6}$$

Divide as indicated. $$\frac{y^{2}+4 y-21}{y^{2}+3 y-28}+\frac{y^{2}+14 y+48}{y^{2}+4 y-32}$$$

A company that manufactures wheelchairs has fixed costs of \(\$ 500,000 .\) The average cost per wheelchair, \(C,\) for the company to manufacture \(x\) wheelchairs per month is modeled by the formula $$C=\frac{400 x+500,000}{x}$$ Use this mathematical model to solve Exercises. How many wheelchairs per month can be produced at an average cost of \(\$ 405\) per wheelchair?

denominators are opposites, or additive inverses. Add or subtract as indicated. Simplify the result, if possible. $$\frac{x-2}{x^{2}-25}-\frac{x-2}{25-x^{2}}$$

Will help you prepare for the material covered in the next section. If \(B=k W,\) find the value of \(k,\) in decimal form, using \(B=5\) and \(W=160\)

Simplify each rational expression. If the rational expression cannot be simplified, so state. $$\frac{x+4}{x^{2}+16}$$

Determine whether statement "makes sense" or "does not make sense" and explain your reasoning. I added 1 to \(\frac{1}{1+\frac{1}{2}}\) and obtained \(\frac{5}{3}\).

Add or subtract as indicated. Simplify the result, if possible. $$\frac{x-5}{x+3}+\frac{x+3}{x-5}$$

Divide as indicated. $$\frac{2 y^{2}-128}{y^{2}+16 y+64}+\frac{y^{2}-6 y-16}{3 y^{2}+30 y+48}$$

denominators are opposites, or additive inverses. Add or subtract as indicated. Simplify the result, if possible. $$\frac{x-8}{x^{2}-16}-\frac{x-8}{16-x^{2}}$$

Will help you prepare for the material covered in the next section. a. If \(y=\frac{k}{x},\) find the value of \(k\) using \(x=8\) and \(y=12\) b. Substitute the value for \(k\) into \(y=\frac{k}{x}\) and write the resulting equation. c. Use the equation from part (b) to find \(y\) when \(x=3\)

Simplify each rational expression. If the rational expression cannot be simplified, so state. $$\frac{x+5}{x^{2}+25}$$

Determine whether statement "makes sense" or "does not make sense" and explain your reasoning. I used the LCD method to simplify $$\frac{3-\frac{6}{x}}{1+\frac{7}{y}}$$ and obtained \(\frac{3-6 y}{1+7 x}\).