Chapter 7

Add or subtract as indicated. Simplify the result, if possible. $$\frac{7}{x-1}-\frac{3}{(x-1)^{2}}$$

Divide as indicated. $$\frac{x^{2}-4}{x} \div \frac{x+2}{x-2}$$

Solve each rational equation. $$\frac{1}{x-4}-\frac{5}{x+2}=\frac{6}{x^{2}-2 x-8}$$

denominators are opposites, or additive inverses. Add or subtract as indicated. Simplify the result, if possible. $$\frac{9 x-1}{7 x-3}+\frac{6 x-2}{3-7 x}$$

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I used \(\frac{a}{d}=\frac{b}{e}\) to show that corresponding sides of similar triangles are proportional, but I could also use \(\frac{a}{b}=\frac{d}{e}\) or \(\frac{d}{a}=\frac{e}{b}\)

Simplify each rational expression. If the rational expression cannot be simplified, so state. $$\frac{3 y^{2}+4 y-4}{6 y^{2}-y-2}$$

Simplify complex rational expression. \(\frac{y^{-1}-(y+2)^{-1}}{2}\)

Add or subtract as indicated. Simplify the result, if possible. $$\frac{5}{x+3}-\frac{2}{(x+3)^{2}}$$

Divide as indicated. $$\frac{x^{2}-4}{x-2}+\frac{x+2}{4 x-8}$$

Solve each rational equation. $$\frac{6}{x+3}-\frac{5}{x-2}=\frac{-20}{x^{2}+x-6}$$

denominators are opposites, or additive inverses. Add or subtract as indicated. Simplify the result, if possible. $$\frac{x^{2}}{x-2}+\frac{4}{2-x}$$

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I can clean my house in 3 hours and my sloppy friend can completely mess it up in 6 hours, so if we both "work" together, the time, \(x,\) it takes to clean the house can be modeled by \(\frac{x}{3}-\frac{x}{6}=1\)

Simplify each rational expression. If the rational expression cannot be simplified, so state. $$\frac{2 x+3}{2 x+5}$$

Simplify complex rational expression. \(\frac{1}{1-\frac{1}{x}}-1\)

Add or subtract as indicated. Simplify the result, if possible. $$\frac{3 y}{4 y-20}+\frac{9 y}{6 y-30}$$

Divide as indicated. $$\left(y^{2}-16\right)+\frac{y^{2}+3 y-4}{y^{2}+4}$$

Solve each rational equation. $$\frac{2}{x+3}-\frac{2 x+3}{x-1}=\frac{6 x-5}{x^{2}+2 x-3}$$

denominators are opposites, or additive inverses. Add or subtract as indicated. Simplify the result, if possible. $$\frac{x^{2}}{x-3}+\frac{9}{3-x}$$

Two skiers begin skiing along a trail at the same time. The faster skier averages 9 miles per hour and the slower skier averages 6 miles per hour. The faster skier completes the trail \(\frac{1}{4}\) hour before the slower skier. How long is the trail?

Simplify each rational expression. If the rational expression cannot be simplified, so state. $$\frac{3 x+7}{3 x+10}$$

Simplify complex rational expression. \(\frac{1}{1-\frac{1}{x+1}}-1\)

Add or subtract as indicated. Simplify the result, if possible. $$\frac{4 y}{5 y-10}+\frac{3 y}{10 y-20}$$

Divide as indicated. $$\left(y^{2}+4 y-5\right)+\frac{y^{2}-25}{y+7}$$

Solve each rational equation. $$\frac{x-3}{x-2}+\frac{x+1}{x+3}=\frac{2 x^{2}-15}{x^{2}+x-6}$$

denominators are opposites, or additive inverses. Add or subtract as indicated. Simplify the result, if possible. $$\frac{y-3}{y^{2}-25}+\frac{y-3}{25-y^{2}}$$

A snowstorm causes a bus driver to decrease the usual average rate along a 60 -mile route by 15 miles per hour. As a result, the bus takes two hours longer than usual to complete the route. At what average rate does the bus usually cover the 60 -mile route?

Simplify each rational expression. If the rational expression cannot be simplified, so state. $$\frac{x^{2}+12 x+36}{x^{2}-36}$$

Simplify complex rational expression. \(\frac{1}{1+\frac{1}{1+\frac{1}{x}}}\)

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

Divide as indicated. $$\frac{y^{2}-y}{15} \div \frac{y-1}{5}$$

Solve or simplify, whichever is appropriate. $$\frac{x^{2}-10}{x^{2}-x-20}=1+\frac{7}{x-5}$$

denominators are opposites, or additive inverses. Add or subtract as indicated. Simplify the result, if possible. $$\frac{y-7}{y^{2}-16}+\frac{7-y}{16-y^{2}}$$

One pipe can fill a swimming pool in 2 hours, a second can fill the pool in 3 hours, and a third pipe can fill the pool in 4 hours. How many minutes, to the nearest minute, would it take to fill the pool with all three pipes operating?

Simplify each rational expression. If the rational expression cannot be simplified, so state. $$\frac{x^{2}-14 x+49}{x^{2}-49}$$

Simplify complex rational expression. \(\frac{1}{1+\frac{1}{1+\frac{1}{2}}}\)

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

Divide as indicated. $$\frac{y^{2}-2 y}{15} \div \frac{y-2}{5}$$

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. $$\frac{6}{x-1}-\frac{5}{1-x}$$

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

The average rate on a round-trip commute having a one-way distance \(d\) is given by the complex rational expression $$\frac{2 d}{\frac{d}{r_{1}}+\frac{d}{r_{2}}},$$ in which \(r_{1}\) and \(r_{2}\) are the average rates on the outgoing and return trips, respectively. Simplify the expression. Then find your average rate if you drive to campus averaging 40 miles per hour and return home on the same route averaging 30 miles per hour.

Add or subtract as indicated. Simplify the result, if possible. $$\frac{2 x+9}{x^{2}-7 x+12}-\frac{2}{x-3}$$

Divide as indicated. $$\frac{4 x^{2}+10}{x-3}+\frac{6 x^{2}+15}{x^{2}-9}$$

Solve or simplify, whichever is appropriate. $$\frac{x^{2}-10}{x^{2}-x-20}-1-\frac{7}{x-5}$$

denominators are opposites, or additive inverses. Add or subtract as indicated. Simplify the result, if possible. $$\frac{10}{x-2}-\frac{6}{2-x}$$

An experienced carpenter can panel a room 3 times faster than an apprentice can. Working together, they can panel the room in 6 hours. How long would it take each person working alone to do the job?

Simplify each rational expression. If the rational expression cannot be simplified, so state. $$\frac{x^{3}+4 x^{2}-3 x-12}{x+4}$$

If two electrical resistors with resistances \(R_{1}\) and \(R_{2}\) are connected in parallel (see the figure), then the total resistance in the circuit is given by the complex rational expression $$\frac{1}{\frac{1}{R_{1}}+\frac{1}{R_{2}}}.$$ Simplify the expression. Then find the total resistance if \(R_{1}=10\) ohms and \(R_{2}=20\) ohms.

Add or subtract as indicated. Simplify the result, if possible. $$\frac{3 x+7}{x^{2}-5 x+6}-\frac{3}{x-3}$$

Divide as indicated. $$\frac{x^{2}+x}{x^{2}-4}+\frac{x^{2}-1}{x^{2}+5 x+6}$$$