Chapter 11

For Exercises 21 to \(32,\) solve for \(y\). $$5 x-y=7$$

Find the LCM of the polynomials. $$\begin{aligned} &x^{2}+7 x+10\\\ &x^{2}-25 \end{aligned}$$

Simplify. $$\frac{x^{2}+8 x+16}{x^{2}-2 x-24}$$

Simplify. $$\frac{1-\frac{2 x-2}{3 x-1}}{x-\frac{4}{3 x-1}}$$

Two oil pipelines can fill a small tank in 30 min. One of the pipelines, working alone, would require 45 min to fill the tank. How long would it take the second pipeline, working alone, to fill the tank?

Which expressions are equivalent to \(\frac{3}{y-5}-\frac{y-2}{y-5} ?\) (I) \(\frac{5-y}{y-5}\) (II) \(\frac{1-y}{y-5}\) (III) \(\frac{5-y}{2 y-10}\) (IV) \(-1\) (V) \(\frac{1-y}{-10}\)

Solve. $$\frac{5}{x+3}=\frac{3}{x-1}$$

For Exercises 21 to \(32,\) solve for \(y\). $$3 x+2 y=6$$

Find the LCM of the polynomials. $$\begin{aligned} &2 x^{2}-7 x+3\\\ &2 x^{2}+x-1 \end{aligned}$$

Simplify. $$\frac{x^{2}-3 x-10}{25-x^{2}}$$

Simplify. $$\frac{x-5+\frac{14}{x+4}}{x+3-\frac{2}{x+4}}$$

A cement mason can construct a retaining wall in 8 h. A second mason requires 12 h to do the same job. After working alone for \(4 \mathrm{h}\), the first mason quits. How long will it take the second mason to complete the wall?

True or false? \(\frac{3}{x-8}+\frac{3}{8-x}=0\)

Solve. $$\frac{2}{3 x-1}=\frac{3}{4 x+1}$$

For Exercises 21 to \(32,\) solve for \(y\). $$2 x+3 y=9$$

Find the LCM of the polynomials. $$\begin{aligned} &3 x^{2}-11 x+6\\\ &3 x^{2}+4 x-4 \end{aligned}$$

Simplify. $$\frac{4-y^{2}}{y^{2}-3 y-10}$$

Simplify. $$\frac{a+4+\frac{5}{a-2}}{a+6+\frac{15}{a-2}}$$

With two reapers operating, a field can be harvested in 1 h. If only the newer reaper is used, the crop can be harvested in \(1.5 \mathrm{h}\). How long would it take to harvest the field using only the older reaper?

Simplify. $$\frac{4}{x}+\frac{5}{y}$$

Solve. $$\frac{5}{3 x-4}=\frac{-3}{1-2 x}$$

For Exercises 21 to \(32,\) solve for \(y\). $$2 x-5 y=10$$

Find the LCM of the polynomials. $$\begin{aligned} &6+x-x^{2}\\\ &x+2\\\ &x-3 \end{aligned}$$

Simplify. $$\frac{2 x^{3}+2 x^{2}-4 x}{x^{3}+2 x^{2}-3 x}$$

Simplify. $$\frac{x+3-\frac{10}{x-6}}{x+2-\frac{20}{x-6}}$$

A manufacturer of prefabricated homes has the company's employees work in teams. Team 1 can erect the Silvercrest model in 15 h. Team 2 can erect the same model in 10 h. How long would it take for Team 1 and Team 2, working together, to erect the Silvercrest model home?

Simplify. $$\frac{7}{a}+\frac{5}{b}$$

Solve. $$\frac{-3}{2 x+5}=\frac{2}{x-1}$$

For Exercises 21 to \(32,\) solve for \(y\). $$5 x-2 y=4$$

How many factors of \(x-3\) are in the LCM of each pair of expressions? a. \(x^{2}+x-12\) and \(x^{2}-9 \quad\) b. \(x^{2}-x-12\) and \(x^{2}+6 x+9 \quad\) c. \(x^{2}+x-12\) and \(x^{2}-6 x+9\)

Simplify. $$\frac{3 x^{3}-12 x}{6 x^{3}-24 x^{2}+24 x}$$

Simplify. $$\frac{x-7+\frac{5}{x-1}}{x-3+\frac{1}{x-1}}$$

One technician can wire a security alarm in \(4 \mathrm{h}\), whereas it takes \(6 \mathrm{h}\) for a second technician to do the same job. After working alone for \(2 \mathrm{h}\), the first technician quits. How long will it take the second technician to complete the wiring?

Simplify. $$\frac{12}{x}-\frac{5}{2 x}$$

Solve. $$\frac{4}{5 y-1}=\frac{2}{2 y-1}$$

For Exercises 21 to \(32,\) solve for \(y\). $$2 x+7 y=14$$

write the fractions in terms of the LCM of the denominators. $$\frac{4}{x}, \frac{3}{x^{2}}$$

Simplify. $$\frac{6 x^{2}-7 x+2}{6 x^{2}+5 x-6}$$

Simplify. $$\frac{y-6+\frac{22}{2 y+3}}{y-5+\frac{11}{2 y+3}}$$

A wallpaper hanger requires 2 h to hang the wallpaper on one wall of a room. A second wallpaper hanger requires 4 h to hang the same amount of wallpaper. The first wallpaper hanger works alone for 1 h and then quits. How long will it take the second hanger, working alone, to finish papering the wall?

Simplify. $$\frac{5}{3 a}-\frac{3}{4 a}$$

Solve. $$\frac{4 x}{x-4}+5=\frac{5 x}{x-4}$$

For Exercises 21 to \(32,\) solve for \(y\). $$6 x-5 y=10$$

Write the fractions in terms of the LCM of the denominators. $$\frac{5}{a b^{2}}, \frac{6}{a b}$$

Simplify. $$\frac{2 n^{2}-9 n+4}{2 n^{2}-5 n-12}$$

Simplify. $$\frac{x+2-\frac{12}{2 x-1}}{x+1-\frac{9}{2 x-1}}$$

A large heating unit and a small heating unit are being used to heat the water in a pool. The large unit, working alone, requires 8 h to heat the pool. After both units have been operating for \(2 \mathrm{h}\), the large unit is turned off. The small unit requires 9 more hours to heat the pool. How long would it take the small unit, working alone, to heat the pool?

Simplify. $$\frac{1}{2 x}-\frac{5}{4 x}+\frac{7}{6 x}$$

Solve. $$\frac{2 x}{x+2}-5=\frac{7 x}{x+2}$$

Write the fractions in terms of the LCM of the denominators. $$\frac{x}{3 y^{2}}, \frac{z}{4 y}$$