Chapter 2

When the equation \(2-\frac{3}{b}=\frac{5}{b+2}\) is solved for \(b\) , the solutions are \(-1\) and \(3 .\) Explain why the number line must be separated into five segments by the numbers \(-2,-1,0,\) and 3 in order to check the solution set of the inequality \(2-\frac{3}{b}>\frac{5}{b+2}\)

Ashley said that \(\frac{(a+2)(a-1)}{(a+3)(a-1)}=\frac{a+2}{a+3}\) for all values of \(a\) except \(a=-3 .\) Do you agree with Ashley? Explain why or why not.

Samantha said that the equation \(\frac{a-2}{a}=\frac{a+2}{2 a}\) in Example 2 could be solved by multiplying both sides of the equation by 2\(a\) . Would Samantha's solution be the same as the solution obtained in Example 2\(?\) Explain why or why not.

Joshua wanted to write this division in simplest form: \(\frac{3}{x-2} \div \frac{4(x-2)}{7} .\) He began by canceling (x-2) in the numerator and denominator and wrote following: $$\frac{3}{x-2} \div \frac{4(x-2)}{7}=\frac{3}{1} \div \frac{4}{7}=\frac{3}{1} \times \frac{7}{4}=\frac{21}{4}$$ Is Joshua's answer correct? Justify your answer.

For what values of \(a\) is \(\left(1-\frac{1}{a}\right) \div\left(1-\frac{1}{a^{2}}\right)=\frac{1-\frac{1}{a}}{1-\frac{1}{a^{2}}}\) undefined? Explain your answer.

If \(\frac{a}{b}=\frac{c}{d},\) then is \(\frac{a}{c}=\frac{b}{d} ?\) Justify your answer.

a. Why is a coin that is worth 25 cents called a quarter? b. Why is the number of minutes in a quarter of an hour different from the number of cents in a quarter of a dollar?

Abby said that \(\frac{3 x}{3 x+4}\) can be reduced to lowest terms by canceling 3\(x\) so that the result is \(\frac{1}{4}\) . Do you agree with Abby? Explain why or why not.

What is the solution set of \(\frac{|x|}{x}<0 ?\) Justify your answer.

Matthew said that \(\frac{a}{b}+\frac{c}{d}=\frac{a d+b c}{b d}\) when \(b \neq 0, d \neq 0 .\) Do you agree with Matthew? Justify your answer.

Brianna said that \(\frac{3}{x-2}=\frac{5}{x+2}\) is a rational equation but \(\frac{x-2}{3}=\frac{x+2}{5}\) is not. Do you agree with Brianna? Explain why or why not.

Gabriel wrote \(\frac{12 x}{5 x+10} \div \frac{4}{5}=\frac{12 x \div 4}{(5 x+10) \div 5}=\frac{3 x}{x+2} .\) Is Gabriel's solution correct? Justify your answer.

Bebe said that since each of the denominators in the complex fraction \(\frac{\frac{d}{4}+\frac{3}{5}}{2-\frac{d^{2}}{2}}\) is a non-zero constant, the fraction is defined for all values of \(d\). Do you agree with Bebe? Explain why or why not.

Write each rational expression in simplest form and list the values of the variables for which the fraction is undefined. If \(\frac{a}{b}=\frac{c}{d},\) then is \(\frac{a+b}{b}=\frac{c+d}{d} ?\) Justify your answer.

Explain the difference between the additive inverse and the multiplicative inverse.

In \(3-14,\) solve and check each inequality. $$ \frac{a}{4}>\frac{a}{2}+6 $$

In \(3-20\) , perform the indicated additions or subtractions and write the result in simplest form. In each case, list any values of the variables for which the fractions are not defined. $$ \frac{x}{3}+\frac{2 x}{3} $$

In \(3-20,\) solve each equation and check. $$ \frac{1}{4} a+8=\frac{1}{2} a $$

In \(3-12,\) multiply and express each product in simplest form. In each case, list any values of the variables for which the fractions are not defined. $$ \frac{2}{3} \times \frac{3}{4} $$

Simplify each complex rational expression. In each case, list any values of the variables for which the fractions are not defined. \(\frac{3}{\frac{3}{4}}\)

Write each ratio in simplest form. \(12 : 8\)

List the values of the variables for which the rational expression is undefined. \(\frac{5 a^{2}}{3 a}\)

In \(3-7,\) write the reciprocal (multiplicative inverse) of each given number. $$ \frac{3}{8} $$

In \(3-14,\) solve and check each inequality. $$ \frac{y-3}{5}<\frac{y+2}{10} $$

In \(3-20\) , perform the indicated additions or subtractions and write the result in simplest form. In each case, list any values of the variables for which the fractions are not defined. $$ \frac{2 x^{2}+1}{5 x}-\frac{7 x^{2}-1}{5 x} $$

In \(3-20,\) solve each equation and check. $$ \frac{3}{4} x=14-x $$

In \(3-12,\) multiply and express each product in simplest form. In each case, list any values of the variables for which the fractions are not defined. $$ \frac{5}{7 a} \cdot \frac{3 a}{20} $$

Simplify each complex rational expression. In each case, list any values of the variables for which the fractions are not defined. \(\frac{\frac{2}{5}}{4}\)

Write each ratio in simplest form. \(21 : 14\)

List the values of the variables for which the rational expression is undefined. \(\frac{-2 d}{6 c}\)

In \(3-7,\) write the reciprocal (multiplicative inverse) of each given number. $$ \frac{7}{12} $$

In \(3-14,\) solve and check each inequality. $$ \frac{3 b-4}{8}<\frac{4 b-3}{4} $$

In \(3-20\) , perform the indicated additions or subtractions and write the result in simplest form. In each case, list any values of the variables for which the fractions are not defined. $$ \frac{x}{7}+\frac{x}{3} $$

In \(3-20,\) solve each equation and check. $$ \frac{x+2}{5}=\frac{x-2}{3} $$

In \(3-12,\) multiply and express each product in simplest form. In each case, list any values of the variables for which the fractions are not defined. $$ \frac{4 y}{5 x} \cdot \frac{x}{8 y} $$

Simplify each complex rational expression. In each case, list any values of the variables for which the fractions are not defined. \(\frac{\frac{7}{8}}{1 \frac{3}{4}}\)

Write each ratio in simplest form. \(3 : 18\)

List the values of the variables for which the rational expression is undefined. \(\frac{a+2}{a b}\)

In \(3-7,\) write the reciprocal (multiplicative inverse) of each given number. $$ \frac{-2}{7} $$

In \(3-14,\) solve and check each inequality. $$ \frac{2-d}{7}>\frac{d-2}{5} $$

In \(3-20\) , perform the indicated additions or subtractions and write the result in simplest form. In each case, list any values of the variables for which the fractions are not defined. $$ \frac{a-1}{5}-\frac{a+1}{4} $$

In \(3-20,\) solve each equation and check. $$ \frac{x}{5}-\frac{x}{10}=7 $$

In \(3-12,\) multiply and express each product in simplest form. In each case, list any values of the variables for which the fractions are not defined. $$ \frac{3 a}{5} \cdot \frac{10}{9 a} $$

Simplify each complex rational expression. In each case, list any values of the variables for which the fractions are not defined. \(\frac{\frac{2}{x}}{\frac{1}{2 x}}\)

Write each ratio in simplest form. \(15 : 75\)

List the values of the variables for which the rational expression is undefined. \(\frac{x-5}{x+5}\)

In \(3-7,\) write the reciprocal (multiplicative inverse) of each given number. $$ 8 $$

In \(3-14,\) solve and check each inequality. $$ \frac{a+1}{4}-2>11-\frac{a}{6} $$

In \(3-20\) , perform the indicated additions or subtractions and write the result in simplest form. In each case, list any values of the variables for which the fractions are not defined. $$ \frac{y+2}{2}+\frac{2 y-3}{3} $$

In \(3-20,\) solve each equation and check. $$ \frac{2 x}{3}+1=\frac{3 x}{4} $$