Chapter 7

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.. $$Divide:$\frac{9 x^{2}-y^{2}+15 x-5 y}{3 x^{2}+x y+5 x} \div \frac{3 x+y}{9 x^{3}+6 x^{2} y+x y^{2}}$$

A drug is injected into a patient and the concentration of the drug in the bloodstream is monitored. The drug's concentration, \(y,\) in milligrams per liter, after \(x\) hours is modeled by $$y=\frac{5 x}{x^{2}+1}$$ The graph of this equation, obtained with a graphing utility, is shown in the figure in a \([0,10,1]\) by \([0,3,1]\) viewing rectangle. Use this information. Use the equation to find the drug's concentration after 3 hours. Then identify the point on the equation's graph that conveys this information.

Perform the indicated operation or operations. Simplify the result, if possible. $$\frac{3}{x^{2}+4 x y+3 y^{2}}-\frac{5}{x^{2}-2 x y-3 y^{2}}+\frac{2}{x^{2}-9 y^{2}}$$

Solve $$2 x+3<3(x-5)$$

Perform the indicated operation or operations. Simplify the result, if possible. $$\frac{5}{x^{2}+3 x y+2 y^{2}}-\frac{7}{x^{2}-x y-2 y^{2}}+\frac{4}{x^{2}-4 y^{2}}$$

Use the GRAPH or TABLE feature of a graphing utility to determine if the multiplication or division has been performed correctly. If the answer is wrong, correct it and then verify your correction using the graphing utility. $$\frac{x^{3}-25 x}{x^{2}-3 x-10} \cdot \frac{x+2}{x}=x+5$$

What is a rational expression? Give an example with your explanation.

Two formulas that approximate the dosage of a drug prescribed for children are Young's rule: \(\quad C=\frac{D A}{A+12}\) and Cowling's rule: \(\quad C=\frac{D(A+1)}{24}\) In each formula, \(A=\) the child's age, in years, \(D=\) an adult dosage, and \(C=\) the proper child's dosage. The formulas apply for ages 2 through \(13,\) inclusive. Use Young's rule to find the difference in a child's dosage for an 8 -year-old child and a 3 -year-old child. Express the answer as a single rational expression in terms of \(D .\) Then describe what your answer means in terms of the variables in the model.

Use the GRAPH or TABLE feature of a graphing utility to determine if the multiplication or division has been performed correctly. If the answer is wrong, correct it and then verify your correction using the graphing utility. $$\frac{x^{2}-9}{x+4} \div \frac{x-3}{x+4}=x-3$$

Explain how to find the number or numbers, if any, for which a rational expression is undefined.

Two formulas that approximate the dosage of a drug prescribed for children are Young's rule: \(\quad C=\frac{D A}{A+12}\) and Cowling's rule: \(\quad C=\frac{D(A+1)}{24}\) In each formula, \(A=\) the child's age, in years, \(D=\) an adult dosage, and \(C=\) the proper child's dosage. The formulas apply for ages 2 through \(13,\) inclusive. Use Young's rule to find the difference in a child's dosage for a 10 -year-old child and a 3 -year-old child. Express the answer as a single rational expression in terms of \(D .\) Then describe what your answer means in terms of the variables in the model.

Use the GRAPH or TABLE feature of a graphing utility to determine if the multiplication or division has been performed correctly. If the answer is wrong, correct it and then verify your correction using the graphing utility. $$(x-5) \div \frac{2 x^{2}-11 x+5}{4 x^{2}-1}=2 x-1$$

Explain how to simplify a rational expression.

Two formulas that approximate the dosage of a drug prescribed for children are Young's rule: \(\quad C=\frac{D A}{A+12}\) and Cowling's rule: \(\quad C=\frac{D(A+1)}{24}\) In each formula, \(A=\) the child's age, in years, \(D=\) an adult dosage, and \(C=\) the proper child's dosage. The formulas apply for ages 2 through \(13,\) inclusive. For a 12 -year-old child, what is the difference in the dosage given by Cowling's rule and Young's rule? Express the answer as a single rational expression in terms of \(D .\) Then describe what your answer means in terms of the variables in the models.

Solve: $$2 x+3<3(x-5)$$.

use the GRAPH or TABLE feature of a graphing utility to determine if the subtraction has been performed correctly. If the answer is wrong, correct it and then verify your correction using the graphing utility. $$\frac{3 x+6}{2}-\frac{x}{2}=x+3$$

Explain how to simplify a rational expression with opposite factors in the numerator and denominator.

Two formulas that approximate the dosage of a drug prescribed for children are Young's rule: \(\quad C=\frac{D A}{A+12}\) and Cowling's rule: \(\quad C=\frac{D(A+1)}{24}\) In each formula, \(A=\) the child's age, in years, \(D=\) an adult dosage, and \(C=\) the proper child's dosage. The formulas apply for ages 2 through \(13,\) inclusive. Use Cowling's rule to find the difference in a child's dosage for a 12 -year- old child and a 10 -year-old child. Express the answer as a single rational expression in terms of \(D .\) Then describe what your answer means in terms of the variables in the model.

Factor $$3 x^{2}-15 x-42$$

use the GRAPH or TABLE feature of a graphing utility to determine if the subtraction has been performed correctly. If the answer is wrong, correct it and then verify your correction using the graphing utility. $$\frac{x^{2}+4 x+3}{x+2}-\frac{5 x+9}{x+2}=x-2, x \neq-2$$

Solve: \(x(2 x+9)=5\)

use the GRAPH or TABLE feature of a graphing utility to determine if the subtraction has been performed correctly. If the answer is wrong, correct it and then verify your correction using the graphing utility. $$\frac{x^{2}-13}{x+4}-\frac{3}{x+4}=x+4, x \neq-4$$

Determine whether each statement “makes sense” or “does not make sense” and explain your reasoning. Simplifying rational expressions is similar to reducing fractions.

Will help you prepare for the materia$l covered in the next section. Subtract: $$\frac{7}{9}-\frac{1}{9}$$

Subtract: \(\frac{13}{15}-\frac{8}{45}\) (Section 1.2, Example 9)

Determine whether each statement “makes sense” or “does not make sense” and explain your reasoning. I cannot simplify rational expressions without knowing how to factor polynomials.

Will help you prepare for the materia$l covered in the next section. Add: $$\frac{2 x}{3}+\frac{x}{3}$$

Factor completely: \(81 x^{4}-1\)

Determine whether each statement “makes sense” or “does not make sense” and explain your reasoning. The rational expressions $$\frac{7}{14 x} and \frac{7}{14+x}$$ an both be simplified by dividing each numerator and each denominator by 7.

Will help you prepare for the materia$l covered in the next section. Simplify: $$\frac{x^{2}-6 x+9}{x^{2}-9}$$

perform the indicated operation. Where possible, reduce the answer to its lowest terms. $$\frac{1}{2}+\frac{2}{3}$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$\frac{3 x+1}{3}=x+1$$

perform the indicated operation. Where possible, reduce the answer to its lowest terms. $$\frac{1}{8}-\frac{5}{6}$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$\frac{x^{2}+3}{3}=x^{2}+1$$

Explain how to find the least common denominator for denominators of \(x^{2}-100\) and \(x^{2}-20 x+100\)

perform the indicated operation. Where possible, reduce the answer to its lowest terms. $$\text { Simplify: } \frac{(y+2) y-2 \cdot 4}{4 y(y+4)}$$

Explain how to add rational expressions that have different denominators. Use \(\frac{3}{x+5}+\frac{7}{x+2}\) in your explanation.

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$\frac{3 x+7}{3 x+10}=\frac{8}{11}$$

Then rewrite the right side of the equation to correct the error that now exists. $$\frac{1}{x}+\frac{2}{5}=\frac{3}{x+5}$$

Write a rational expression that cannot be simplified.

Write a rational expression that is undefined for \(x=-4\).

Relate the dosage of a drug prescribed for children to the child's age. Describe another factor that might be used when determining a child's dosage. Is this factor more or less important than age? Explain why.

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. It takes me more steps to find \(\frac{2}{x+5}+\frac{7}{x-3}\) than it does to find \(\frac{2 x^{3}}{x+5}+\frac{7 x^{3}}{x+5}\)

Use the GRAPH or TABLE feature of a graphing utility to determine if the rational expression has been correctly simplified. If the simplification is wrong, correct it and then verify your answer using the graphing utility. $$\frac{3 x+15}{x+5}=3, x \neq-5$$

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. The reason I can rewrite rational expressions with a common denominator is that 1 is the multiplicative identity.

Use the GRAPH or TABLE feature of a graphing utility to determine if the rational expression has been correctly simplified. If the simplification is wrong, correct it and then verify your answer using the graphing utility. $$\frac{2 x^{2}-x-1}{x-1}=2 x^{2}-1, x \neq 1$$

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. The fastest way for me to find \(\frac{4}{x-5}+\frac{9}{5-x}\) is by using \((x-5)(5-x)\) as the \(\mathrm{LCD}\)

Use the GRAPH or TABLE feature of a graphing utility to determine if the rational expression has been correctly simplified. If the simplification is wrong, correct it and then verify your answer using the graphing utility. $$\frac{x^{2}-x}{x}=x^{2}-1, x \neq 0$$

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. After adding rational expressions with different denominators, I factored the numerator and found no common factors in the numerator and denominator, so my final answer is incorrect if I leave the numerator in factored form.

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$x-\frac{1}{5}=\frac{4}{5} x$$