Chapter 4

Set up an algebraic equation and solve each problem. The total value of a house and a lot is \(\$ 168,000\). If the ratio of the value of the house to the value of the lot is 7 to 1 , find the value of the house.

For problems \(53-64\), use synthetic division to determine the quotient and remainder. $$ \left(x^{2}+9 x+18\right) \div(x+3) $$

Simplify each complex fraction. $$ \frac{4+\frac{6}{n-1}}{7-\frac{4}{n-1}} $$

Add or subtract the rational expressions as indicated. Be sure to express your answers in simplest form. $$ \frac{-2}{n-6}-\frac{6}{2 n+3} $$

For Problems 51-58, simplify each rational expression. You will need to use factoring by grouping. \(\frac{x^{2}-2 x+a x-2 a}{x^{2}-2 x+3 a x-6 a}\)

Rod agreed to mow a vacant lot for $$\$ 12$$. It took him an hour longer than he had anticipated, so he earned $$\$ 1$$ per hour less than he had originally calculated. How long had he anticipated that it would take him to mow the lot?

Set up an algebraic equation and solve each problem. The sum of two numbers is 90 . If the larger is divided by the smaller, the quotient is 10 , and the remainder is 2 . Find the numbers.

For problems \(53-64\), use synthetic division to determine the quotient and remainder. $$ \left(x^{2}+2 x-10\right) \div(x-4) $$

Simplify each complex fraction. $$ \frac{5-\frac{2}{n-3}}{4-\frac{1}{n-3}} $$

Add or subtract the rational expressions as indicated. Be sure to express your answers in simplest form. $$ \frac{-1}{x+4}+\frac{4}{7 x-1} $$

For Problems 51-58, simplify each rational expression. You will need to use factoring by grouping. \(\frac{5 x^{2}+5 x+3 x+3}{5 x^{2}+3 x-30 x-18}\)

Last week Al bought some golf balls for $$\$ 20$$. The next day they were on sale for $$\$ 0.50$$ per ball less, and he bought \(\$ 22.50\) worth of balls. If he purchased 5 more balls on the second day than on the first day, how many did he buy each day and at what price per ball?

Set up an algebraic equation and solve each problem. What number must be added to the numerator and denominator of \(\frac{2}{5}\) to produce a rational number that is equivalent to \(\frac{7}{8} ?\)

For problems \(53-64\), use synthetic division to determine the quotient and remainder. $$ \left(x^{2}-10 x+15\right) \div(x-8) $$

Simplify each complex fraction. $$ \frac{\frac{3}{n-5}-2}{1-\frac{4}{n-5}} $$

Add or subtract the rational expressions as indicated. Be sure to express your answers in simplest form. $$ \frac{-3}{4 x+3}+\frac{5}{2 x-5} $$

For Problems 51-58, simplify each rational expression. You will need to use factoring by grouping. \(\frac{x^{2}+3 x+4 x+12}{2 x^{2}+6 x-x-3}\)

Debbie rode her bicycle out into the country for a distance of 24 miles. On the way back, she took a much shorter route of 12 miles and made the return trip in one-half hour less time. If her rate out into the country was 4 miles per hour greater than her rate on the return trip, find both rates.

Set up an algebraic equation and solve each problem. A 20-foot board is to be cut into two pieces whose lengths are in the ratio of 7 to 3 . Find the lengths of the two pieces.

Use synthetic division to determine the quotient and remainder. $$ \left(x^{3}-2 x^{2}-x+2\right) \div(x-2) $$

Simplify each complex fraction. $$ \frac{\frac{-1}{y-2}+\frac{5}{x}}{\frac{3}{x}-\frac{4}{x y-2 x}} $$

Add or subtract the rational expressions as indicated. Be sure to express your answers in simplest form. $$ \frac{7}{3 x-5}-\frac{5}{2 x+7} $$

For Problems 51-58, simplify each rational expression. You will need to use factoring by grouping. \(\frac{2 s t-30-12 s+5 t}{3 s t-6-18 s+t}\)

Felipe jogs for 10 miles and then walks another 10 miles. He jogs \(2 \frac{1}{2}\) miles per hour faster than he walks, and the entire distance of 20 miles takes 6 hours. Find the rate at which he walks and the rate at which he jogs.

Set up an algebraic equation and solve each problem. An inheritance of \(\$ 300,000\) is to be divided between a son and the local heart fund in the ratio of 3 to 1 . How much money will the son receive?

Use synthetic division to determine the quotient and remainder. $$ \left(x^{3}-5 x^{2}+2 x+8\right) \div(x+1) $$

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

Add or subtract the rational expressions as indicated. Be sure to express your answers in simplest form. $$ \frac{5}{x-1}-\frac{3}{2 x-3} $$

For Problems 51-58, simplify each rational expression. You will need to use factoring by grouping. \(\frac{n r-6-3 n+2 r}{n r+10+2 r+5 n}\)

Why is it important to consider more than one way to do a problem?

Set up an algebraic equation and solve each problem. Suppose that in a certain precinct, 1150 people voted in the last presidential election. If the ratio of female voters to male voters was 3 to 2 , how many females and how many males voted?

Use synthetic division to determine the quotient and remainder. $$ \left(x^{3}-7 x-6\right) \div(x+2) $$

Simplify each complex fraction. $$ \frac{\frac{2}{x-3}-\frac{3}{x+3}}{\frac{5}{x^{2}-9}-\frac{2}{x-3}} $$

Add or subtract the rational expressions as indicated. Be sure to express your answers in simplest form. $$ \frac{5}{3 x-2}+\frac{6}{4 x+5} $$

For Problems 59-68, simplify each rational expression. You may want to refer to Example 12 of this section. \(\frac{5 x-7}{7-5 x}\)

Set up an algebraic equation and solve each problem. The perimeter of a rectangle is 114 centimeters. If the ratio of its width to its length is 7 to 12 , find the dimensions of the rectangle.

Use synthetic division to determine the quotient and remainder. $$ \left(x^{3}+6 x^{2}-5 x-1\right) \div(x-1) $$

Simplify each complex fraction. $$ \frac{\frac{2}{x-y}+\frac{3}{x+y}}{\frac{5}{x+y}-\frac{1}{x^{2}-y^{2}}} $$

Add or subtract the rational expressions as indicated. Be sure to express your answers in simplest form. $$ \frac{3}{2 x+1}+\frac{2}{3 x+4} $$

For Problems 59-68, simplify each rational expression. You may want to refer to Example 12 of this section. \(\frac{4 a-9}{9-4 a}\)

Use synthetic division to determine the quotient and remainder. $$ \left(2 x^{3}-5 x^{2}-4 x+6\right) \div(x-2) $$

Simplify each complex fraction. $$ \frac{3 a}{2-\frac{1}{a}}-1 $$

Add or subtract the rational expressions as indicated. Be sure to express your answers in simplest form. $$ \frac{3 x}{2 x+5}+1 $$

For Problems 59-68, simplify each rational expression. You may want to refer to Example 12 of this section. \(\frac{n^{2}-49}{7-n}\)

Use synthetic division to determine the quotient and remainder. $$ \left(3 x^{4}-x^{3}+2 x^{2}-7 x-1\right) \div(x+1) $$

Simplify each complex fraction. $$ \frac{a}{\frac{1}{a}+4}+1 $$

For Problems 59-68, simplify each rational expression. You may want to refer to Example 12 of this section. \(\frac{9-y}{y^{2}-81}\)

How can you tell by inspection that the equation \(\frac{x}{x+2}=\frac{-2}{x+2}\) has no solution?

Use synthetic division to determine the quotient and remainder. $$ \left(x^{4}+4 x^{3}-7 x-1\right) \div(x-3) $$

Simplify each complex fraction. $$ 2-\frac{x}{3-\frac{2}{x}} $$