Chapter 7

Simplify each rational expression. $$\frac{x^{3}-8}{x^{2}+2 x-8}$$

Explain how to divide rational expressions.

Solve each rational equation. $$\frac{x+1}{2 x^{2}-11 x+5}=\frac{x-7}{2 x^{2}+9 x-5}-\frac{2 x-6}{x^{2}-25}$$

Describe two similarities between the following problems: $$ \frac{3}{8}+\frac{1}{8} \text { and } \frac{x}{x^{2}-1}+\frac{1}{x^{2}-1} $$

Simplify each rational expression. $$\frac{9-y^{2}}{y^{2}-3(2 y-3)}$$

Add or subtract as indicated. Simplify the result, if possible. $$\frac{x-1}{x}+\frac{y+1}{y}$$

In dividing polynomials $$2\(\frac{P}{Q} \div \frac{R}{S}$$ why is it necessary to state that polynomial \)R\( is not equal to \)0 ?$

Solve each rational equation. $$\left(\frac{x+1}{x+7}\right)^{2} \div\left(\frac{x+1}{x+7}\right)^{4}=0$$

Explain how to add rational expressions when denominators are opposites. Use an example to support your explanation.

Simplify each rational expression. $$\frac{16-y^{2}}{y(y-8)+16}$$

Add or subtract as indicated. Simplify the result, if possible. $$\frac{x+2}{y}+\frac{y-2}{x}$$

Find \(b\) so that the solution of \(\frac{7 x+4}{b}+13=x\) is \(-6.\)

Simplify each rational expression. $$\frac{x y+2 y+3 x+6}{x^{2}+5 x+6}$$

Add or subtract as indicated. Simplify the result, if possible. $$\frac{3 x}{x^{2}-y^{2}}-\frac{2}{y-x}$$

Use a graphing utility to solve each rational equation. Graph each side of the equation in the given viewing rectangle. The first coordinate of each point of intersection is a solution. Check by direct substitution. $$\begin{aligned} &\frac{x}{2}+\frac{x}{4}=6\\\ &[-5,10,1] \text { by }[-5,10,1] \end{aligned}$$

After adding \(\frac{3 x+1}{4}\) and \(\frac{x+2}{4},\) I simplified the sum by dividing the numerator and the denominator by 4 I use similar procedures to find each of the following sums: $$ \frac{3}{8}+\frac{1}{8} \text { and } \frac{x}{x^{2}-1}+\frac{1}{x^{2}-1} $$

Simplify each rational expression. $$\frac{x y+4 y-7 x-28}{x^{2}+11 x+28}$$

Add or subtract as indicated. Simplify the result, if possible. $$\frac{7 x}{x^{2}-y^{2}}-\frac{3}{y-x}$$

When performing the division. $$\frac{7 x}{x+3} \div \frac{(x+3)^{2}}{x-5}$$ I began by dividing the numerator and the denominator by the common factor, \(x+3\)

Use a graphing utility to solve each rational equation. Graph each side of the equation in the given viewing rectangle. The first coordinate of each point of intersection is a solution. Check by direct substitution. $$\begin{aligned} &\frac{50}{x}=2 x\\\ &[-10,10,1] \text { by }[-20,20,2] \end{aligned}$$

Simplify each rational expression. $$\frac{8 x^{2}+4 x+2}{1-8 x^{3}}$$

Perform the indicated operation or operations. Simplify the result, if possible. $$\frac{x+6}{x^{2}-4}-\frac{x+3}{x+2}+\frac{x-3}{x-2}$$

The quotient $$\frac{x+2}{x-5}+\frac{x-4}{x+3}$$ is undefined for \(x=5, x=-3,\) and $x=4$$

determine whether each statement “makes sense” or “does not make sense” and explain your reasoning. I added \(\frac{5}{x-7}\) and \(\frac{3}{7-x}\) by first multiplying the second rational expression by \(-1\)

Simplify each rational expression. $$\frac{x^{3}-3 x^{2}+9 x}{x^{3}+27}$$

Perform the indicated operation or operations. Simplify the result, if possible. $$\frac{x+8}{x^{2}-9}-\frac{x+2}{x+3}+\frac{x-2}{x-3}$$

Factor completely: \(x^{4}+2 x^{3}-3 x-6 .\) (Section 6.1 Example 8).

The rational expression $$\frac{130 x}{100-x}$$ describes the cost, in millions of dollars, to inoculate \(x\) percent of the population against a particular strain of flu. a. Evaluate the expression for \(x=40, x=80,\) and \(x=90\) Describe the meaning of each evaluation in terms of percentage inoculated and cost. b. For what value of \(x\) is the expression undefined? c. What happens to the cost as \(x\) approaches \(100 \% ?\) How can you interpret this observation?

Perform the indicated operation or operations. Simplify the result, if possible. $$\frac{5}{x^{2}-25}+\frac{4}{x^{2}-11 x+30}-\frac{3}{x^{2}-x-30}$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$\frac{4}{x} \div \frac{x-2}{x}=\frac{4}{x-2}\( if \)x \neq 0\( and \)x \neq 2$$

Simplify: \(\left(3 x^{2}\right)\left(-4 x^{-10}\right) .\) (Section 5.7, Example 3)

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$\frac{x-5}{6} \cdot \frac{3}{5-x}=\frac{1}{2}\( for any value of \)x$ except 5$$

The rational expression $$\frac{60,000 x}{100-x}$$ describes the cost, in dollars, to remove \(x\) percent of the air pollutants in the smokestack emissions of a utility company that burns coal to generate electricity. a. Evaluate the expression for \(x=20, x=50,\) and \(x=80\) Describe the meaning of each evaluation in terms of percentage of pollutants removed and cost. b. For what value of \(x\) is the expression undefined? c. What happens to the cost as \(x\) approaches \(100 \% ?\) How can you interpret this observation?

Perform the indicated operation or operations. Simplify the result, if possible. $$\frac{3}{x^{2}-49}+\frac{2}{x^{2}-15 x+56}-\frac{5}{x^{2}-x-56}$$

Simplify: \(\quad-5[4(x-2)-3] .\) (Section 1.8 , Example 11)

determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The difference between two rational expressions with the same denominator can always be simplified.

Doctors use the rational expression $$\frac{D A}{A+12}$$ to determine the dosage of a drug prescribed for children. In this expression, \(A=\) the child's age and \(D=\) the adult dosage. Use the expression. If the normal adult dosage of medication is 1000 milligrams, what dosage should an 8 -year-old child receive?

Perform the indicated operation or operations. Simplify the result, if possible. $$\frac{x+6}{x^{3}-27}-\frac{x}{x^{3}+3 x^{2}+9 x}$$

Will help you prepare for the material covered in the next section. Solve: \(\frac{15}{8+x}=\frac{9}{8-x}\)

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.. $$Find the missing polynomials: $\quad-\frac{3 x-12}{2 x}=\frac{3}{2}$$

Doctors use the rational expression $$\frac{D A}{A+12}$$ to determine the dosage of a drug prescribed for children. In this expression, \(A=\) the child's age and \(D=\) the adult dosage. Use the expression. If the normal adult dosage of medication is 1000 milligrams, what dosage should a 4-year-old child receive?

Perform the indicated operation or operations. Simplify the result, if possible. $$\frac{x+8}{x^{3}-8}-\frac{x}{x^{3}+2 x^{2}+4 x}$$

Will help you prepare for the material covered in the next section. If you can complete a job in 5 hours, what fractional part of the job can you complete in 1 hour? in 3 hours? in \(x\) hours?

perform the indicated operations. Simplify the result, if possible. $$\left(\frac{3 x-1}{x^{2}+5 x-6}-\frac{2 x-7}{x^{2}+5 x-6}\right) \div \frac{x+2}{x^{2}-1}$$

A company that manufactures bicycles has costs given by the equation $$ C=\frac{100 x+100,000}{x} $$ in which \(x\) is the number of bicycles manufactured and \(C\) is the cost to manufacture each bicycle. a. Find the cost per bicycle when manufacturing 500 bicycles. b. Find the cost per bicycle when manufacturing 4000 bicycles. c. Does the cost per bicycle increase or decrease as more bicycles are manufactured? Explain why this happens.

Perform the indicated operation or operations. Simplify the result, if possible. $$\frac{9 y+3}{y^{2}-y-6}+\frac{y}{3-y}+\frac{y-1}{y+2}$$

Will help you prepare for the material covered in the next section. If you can complete a job in 5 hours, what fractional part of the job can you complete in 1 hour? in 3 hours? in \(x\) hours?

perform the indicated operations. Simplify the result, if possible. $$\left(\frac{3 x^{2}-4 x+4}{3 x^{2}+7 x+2}-\frac{10 x+9}{3 x^{2}+7 x+2}\right) \div \frac{x-5}{x^{2}-4}$$

A company that manufactures small canoes has costs given by the equation $$ C=\frac{20 x+20,000}{x} $$ in which \(x\) is the number of canoes manufactured and \(C\) is the cost to manufacture each canoe. a. Find the cost per canoe when manufacturing 100 canoes. b. Find the cost per canoe when manufacturing \(10,000\) canoes. c. Does the cost per canoe increase or decrease as more canoes are manufactured? Explain why this happens.

Perform the indicated operation or operations. Simplify the result, if possible. $$\frac{7 y-2}{y^{2}-y-12}+\frac{2 y}{4-y}+\frac{y+1}{y+3}$$