Chapter 11

What does it mean for two quantities to vary directly?

Complete: To add or subtract rational expressions with like denominators, add or subtract their numerators, and write the result over the \(?\)

Explain what is meant by the least common denominator of two rational expressions.

What are two methods of solving rational equations?

Identify the extremes and the means of the proportion. a. \(\frac{3}{4}=\frac{9}{12}\) b. \(\frac{9}{12}=\frac{3}{4}\)

Define rational number. Which of the following are rational numbers? $$5, \frac{2}{3}, \frac{-17}{2}, \sqrt{3}, 1.45,0, \pi$$

What does it mean for two quantities to vary inversely?

Add. Simplify your answer. $$ \frac{1}{3 x}+\frac{5}{3 x} $$

Find and correct the error. $$ \begin{aligned} \frac{3}{x-1}-\frac{2}{x} &=\frac{3 x}{x(x-1)}-\frac{2(x-1)}{x(x-1)} \\ &=\frac{3 x}{x(x-1)}-\frac{2 x-2}{x(x-1)} \\ &=\frac{3 x-2 x-2}{x(x-1)}=\frac{x-2}{x(x-1)} \end{aligned} $$

Which method is limited to solving equations in which each side is a single rational expression?

Solve the proportion. Check your solution. $$ \frac{2}{x}=\frac{16}{40} $$

Define rational expression. Give an example of a rational expression.

Add. Simplify your answer. $$ \frac{8 y}{y+3}+\frac{10-3 y}{y+3} $$

In Exercises 3–6, simplify the expression. $$ \frac{x}{12}+\frac{x}{4} $$

Find the least common denominator. \(\frac{1}{x}, \frac{x}{3}, \frac{2}{3 x}\)

Solve the proportion. Check your solution. $$ \frac{72}{96}=\frac{x}{4} $$

Simplify the expression. $$\frac{3 x}{8 x^{2}} \cdot \frac{4 x^{3}}{3 x^{4}}$$

Define the simplest form of a rational expression. Give an example of a rational expression in simplest form.

Add. Simplify your answer. $$ \frac{x}{x^{2}-9}+\frac{3 x+1}{x^{2}-9} $$

In Exercises 3–6, simplify the expression. $$ \frac{3}{10 x}-\frac{1}{4 x^{2}} $$

Find the least common denominator. \(\frac{3}{4 x}, \frac{1}{6 x^{2}}, \frac{1}{8 x^{2}}\)

Solve the proportion. Check your solution. $$ \frac{x}{3}=\frac{2}{7} $$

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

Simplify the expression. If not possible, write already in simplest form. $$ \frac{28 y}{4} $$

Subtract. Simplify your answer. $$ \frac{8}{3 r}-\frac{1}{3 r} $$

In Exercises 3–6, simplify the expression. $$ \frac{x+6}{x+1}-\frac{4}{2 x+3} $$

Find the least common denominator. \(\frac{5}{x}, \frac{2}{3 x^{2}}, \frac{1}{x^{3}}\)

Solve the proportion. Check your solution. $$ \frac{4}{x+1}=\frac{7}{2} $$

Simplify the expression. $$\frac{x}{x^{2}-25} \cdot \frac{x-5}{x+5}$$

Simplify the expression. If not possible, write already in simplest form. $$ \frac{16}{128 c} $$

Does the equation model direct variation, inverse variation, or neither? $$ x=\frac{4}{y} $$

Subtract. Simplify your answer. $$ \frac{12 k}{k^{2}}-\frac{3 k+7}{k^{2}} $$

In Exercises 3–6, simplify the expression. $$ \frac{x-2}{2 x-10}+\frac{x+3}{x-5} $$

Solve the equation using the cross product property. Remember to check your solutions. \(\frac{3}{x}=\frac{x}{12}\)

Solve the proportion. Check your solution. $$ \frac{2}{2 x+1}=\frac{1}{5} $$

Simplify the expression. $$\frac{3 x}{x^{2}-2 x-15} \cdot(x+3)$$

Simplify the expression. If not possible, write already in simplest form. $$ \frac{12 x^{2}}{6 x} $$

Does the equation model direct variation, inverse variation, or neither? $$ y=7 x-2 $$

Subtract. Simplify your answer. $$ \frac{c+1}{c^{2}-4}-\frac{c+6}{c^{2}-4} $$

You can use \(x-3\) as the LCD when finding the sum \(\frac{5}{x-3}+\frac{2}{3-x} .\) What number can you multiply the numerator and the denominator of the second fraction by to get an equivalent fraction with \(x-3\) as the new denominator?

Solve the equation using the cross product property. Remember to check your solutions. \(\frac{x}{x+2}=\frac{3}{x-2}\)

Solve the proportion. Check your solution. $$ \frac{x-2}{x}=\frac{2}{3} $$

Simplify the expression. $$\frac{x}{8-2 x} \div \frac{2 x}{4-x}$$

Simplify the expression. If not possible, write already in simplest form. $$ \frac{a-8}{4} $$

Does the equation model direct variation, inverse variation, or neither? $$ x=12 y $$

Add or subtract, then factor and simplify. $$ \frac{5 x}{x+4}+\frac{20}{4+x} $$

Find the least common denominator of the pair of rational expressions. $$ \frac{1}{3 x}, \frac{1}{9 x^{3}} $$

Solve the equation using the cross product property. Remember to check your solutions. \(\frac{3}{u+2}=\frac{1}{u-2}\)

Determine whether the equation follows from \(\frac{a}{b}=\frac{c}{d}\). $$ a d=b c $$

Simplify the expression. $$\frac{4 x^{2}-25}{4 x} \div(2 x-5)$$