Chapter 11

Solve the equation. Check your solutions. $$ \frac{2}{3 t}=\frac{t-1}{t} $$

Write the product in simplest form. $$3 z^{2}+10 z+3 \cdot \frac{z+3}{3 z^{2}+4 z+1}$$

Simplify the expression. If not possible, write already in simplest form. $$\frac{x^{2}+8 x+16}{3 x+12}$$

FACTORING AFTER ADDING OR SUBTRACTING. Simplify the expression. $$ \frac{y^{2}}{y^{2}-3 y-28}-\frac{12-y}{y^{2}-3 y-28} $$

In Exercises \(31-33,\) state whether the variables model direct variation, inverse variation, or neither. BASE AND HEIGHT The area \(B\) of the base and the height \(h\) of a prism with a volume of 10 cubic units are related by the equation \(B h=10\)

Solve the equation by multiplying each side by the least common denominator. Check your solutions. \(\frac{5}{x+1}-\frac{7}{x+1}=\frac{12}{x}\)

Write the difference in simplest form. $$ \frac{x-1}{6 x^{2}}-\frac{2}{3 x} $$

Solve the equation. Check your solutions. $$ \frac{x}{2}=\frac{5}{x+3} $$

Write the quotient in simplest form. $$\frac{25 x^{2}}{10 x} \div \frac{5 x}{10 x}$$

Simplify the expression. If not possible, write already in simplest form. $$\frac{x^{2}+x-20}{x^{2}+2 x-15}$$

Find and correct the error. $$ \begin{aligned} &\frac{3 n^{2}}{n^{2}-144}-\frac{36 n}{n^{2}-144}=\frac{3 n^{2}-36 n}{n^{2}-144}\\\ &=\frac{3 n(n-12)}{(n-12)(n-12)}=\frac{3 n}{n-12} \end{aligned} $$

In Exercises \(31-33,\) state whether the variables model direct variation, inverse variation, or neither. MASS AND VOLUME The mass \(m\) and the volume \(V\) of a substance are related by the equation \(2 V=m,\) where 2 is the density of the substance.

Solve the equation by multiplying each side by the least common denominator. Check your solutions. \(\frac{5}{3}+\frac{250}{9 r}=\frac{r}{9}\)

Write the difference in simplest form. $$ \frac{5 c}{15}-\frac{2+c}{25 c} $$

Solve the equation. Check your solutions. $$ \frac{x-3}{18}=\frac{3}{x} $$

Write the quotient in simplest form. $$\frac{16 x^{2}}{8 x} \div \frac{4 x^{2}}{16 x}$$

Simplify the expression. If not possible, write already in simplest form. $$\frac{x^{3}+9 x^{2}+14 x}{x^{2}-4}$$

Find and correct the error. $$ \begin{aligned} &\frac{x+2}{y+3}+\frac{y-4}{y+3}=\frac{(y+2)(y-4)}{(y+3)^{2}}\\\ &=\frac{y^{2}-2 y-8}{y^{2}+6 y+9} \end{aligned} $$

In Exercises \(31-33,\) state whether the variables model direct variation, inverse variation, or neither. HOURS AND PAY RATE The number of hours \(h\) that you must work to earn \(\$ 480\) and your hourly rate of pay \(p\) are related by the equation \(p h=480\)

Factor first, then solve the equation. Check your solutions. \(\frac{2}{y-2}+\frac{1}{y+2}=\frac{4}{y^{2}-4}\)

Write the difference in simplest form. $$ \frac{2 x-1}{3 x}-\frac{1}{11} $$

Solve the equation. Check your solutions. $$ \frac{-2}{a-7}=\frac{a}{5} $$

Write the quotient in simplest form. $$\frac{3 x^{2}}{10} \div \frac{9 x^{3}}{25}$$

Simplify the expression. If not possible, write already in simplest form. $$\frac{x^{3}-x}{x^{3}+5 x^{2}-6 x}$$

Simplify the expression. $$ \frac{11 x-5}{2 x+5}+\frac{11 x+12}{2 x+5}+\frac{3 x-100}{2 x+5} $$

Factor first, then solve the equation. Check your solutions. \(\frac{3}{x+1}-\frac{1}{x-2}=\frac{1}{x^{2}-x-2}\)

Simplify the expression. $$ \frac{2}{x+1}+\frac{3}{x-2} $$

Solve the equation. Check your solutions. $$ \frac{x-3}{x}=\frac{x}{x+6} $$

Write the quotient in simplest form. $$\frac{x}{x+2} \div \frac{x+5}{x+2}$$

Simplify the expression if possible. $$ \frac{x^{2}-9}{x^{2}-5 x-6} $$

Simplify the expression. $$ \frac{4+x}{x-9}+\frac{6+x}{x-9}-\frac{1-x}{x-9} $$

In Exercises \(35-37,\) use the following information. When a person walks, the pressure on each boot sole varies inversely with the area of the sole. Denise is walking through deep snow, wearing boots that have a sole area of 29 square inches each. The pressure on the sole is 4 pounds per square inch when she stands on one foot. Use unit analysis to explain why the constant of variation is Denise's weight. How much does she weigh?

Factor first, then solve the equation. Check your solutions. \(\frac{3}{x-1}+\frac{10}{x^{2}-2 x+1}=4\)

Simplify the expression. $$ \frac{x}{x-10}+\frac{x+4}{x+6} $$

Solve the equation. Check your solutions. $$ \frac{9-x}{x+4}=\frac{5}{2 x} $$

Write the quotient in simplest form. $$\frac{2(x+2)}{5(x-3)} \div \frac{4(x-2)}{5 x-15}$$

Simplify the expression if possible. $$ \frac{2 x^{2}+11 x-6}{x+6} $$

Simplify the expression. $$ \frac{c-15}{2 c+6}-\frac{2 c}{2 c+6}+\frac{12}{2 c+6} $$

Factor first, then solve the equation. Check your solutions. \(\frac{x}{x+3}+\frac{1}{x-1}=\frac{4}{x^{2}+2 x-3}\)

Simplify the expression. $$ \frac{x-3}{x+3}+\frac{x+9}{x-3} $$

Assume that a 15 -meter-wide site is representative of a larger 60 -meter-wide site. If an archaeologist excavates the 15 -meter-wide site and finds 30 clay pots, estimate the number of clay pots in the larger 60 -meter-wide site. Assume that both sites are the same length.

Write the quotient in simplest form. $$\frac{x}{x-2} \div \frac{2 x-2}{x^{2}-3 x+2}$$

Simplify the expression if possible. $$ \frac{121-x^{2}}{x^{2}+15 x+44} $$

Simplify the expression. $$ \frac{2 x}{x^{2}-9}-\frac{4 x+2}{x^{2}-9}-\frac{4}{x^{2}-9} $$

In Exercises \(35-37,\) use the following information. When a person walks, the pressure on each boot sole varies inversely with the area of the sole. Denise is walking through deep snow, wearing boots that have a sole area of 29 square inches each. The pressure on the sole is 4 pounds per square inch when she stands on one foot. If Denise wears snowshoes, each with an area of 319 square inches, what is the pressure on the snowshoe when she stands on one foot?

Factor first, then solve the equation. Check your solutions. \(\frac{2}{x-1}-\frac{x}{x+3}=\frac{6}{x^{2}+2 x-3}\)

Simplify the expression. $$ \frac{x+8}{3 x-1}+\frac{x+3}{x+1} $$

The ratio of the sculpture of John Wesley Dobbs' head to actual size is about 10 to \(1 .\) Suppose that his head was 9 inches high and \(6 \frac{1}{2}\) inches wide. Estimate the height and width of the sculpture. Write the answer in feet.

Write the quotient in simplest form. $$\frac{x}{x+6} \div \frac{x+3}{x^{2}-36}$$

Simplify the expression if possible. $$ \frac{1-x}{x^{2}-x} $$