Chapter 7

Add or subtract as indicated. Simplify the result, if possible. $$\frac{x-7}{x+4}+\frac{x+4}{x-7}$$

Divide as indicated. $$\frac{3 y+12}{y^{2}+3 y} \div \frac{y^{2}+y-12}{9 y-y^{3}}$$

denominators are opposites, or additive inverses. Add or subtract as indicated. Simplify the result, if possible. $$\frac{x}{x-y}+\frac{y}{y-x}$$

Simplify each rational expression. If the rational expression cannot be simplified, so state. $$\frac{x-5}{5-x}$$

Add or subtract as indicated. Simplify the result, if possible. $$\frac{5}{2 y^{2}-2 y}-\frac{3}{2 y-2}$$

Divide as indicated. $$\frac{2 x+2 y}{3} \div \frac{x^{2}-y^{2}}{x-y}$$

We have seen that Young's rule $$C=\frac{D A}{A+12}$$ can be used to approximate the dosage of a drug prescribed for children. In this formula, \(A=\) the child's age, in years, \(D=\)an adult dosage, and \(C=\)the proper child's dosage. Use this formula to solve Exercises. When the adult dosage is 1000 milligrams, a child is given 300 milligrams. What is that child's age? Round to the nearest year.

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

Simplify each rational expression. If the rational expression cannot be simplified, so state. $$\frac{x-7}{7-x}$$

Add or subtract as indicated. Simplify the result, if possible. $$\frac{7}{5 y^{2}-5 y}-\frac{2}{5 y-5}$$

Divide as indicated. $$\frac{5 x+5 y}{7}+\frac{x^{2}-y^{2}}{x-y}$$

We have seen that Young's rule $$C=\frac{D A}{A+12}$$ can be used to approximate the dosage of a drug prescribed for children. In this formula, \(A=\) the child's age, in years, \(D=\)an adult dosage, and \(C=\)the proper child's dosage. Use this formula to solve Exercises. When the adult dosage is 1000 milligrams, a child is given 500 milligrams. What is that child's age?

denominators are opposites, or additive inverses. Add or subtract as indicated. Simplify the result, if possible. $$\frac{2 x}{x^{2}-y^{2}}+\frac{2 y}{y^{2}-x^{2}}$$

Simplify each rational expression. If the rational expression cannot be simplified, so state. $$\frac{2 x-3}{3-2 x}$$

Determine whether statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. \(\frac{y-\frac{1}{2}}{y+\frac{3}{4}}=\frac{4 y-2}{4 y+3}\) for any value of \(y\) except \(-\frac{3}{4}\)

Add or subtract as indicated. Simplify the result, if possible. $$\frac{4 x+3}{x^{2}-9}-\frac{x+1}{x-3}$$

Divide as indicated. $$\frac{x^{2}-y^{2}}{8 x^{2}-16 x y+8 y^{2}} \div \frac{4 x-4 y}{x+y}$$

denominators are opposites, or additive inverses. Add or subtract as indicated. Simplify the result, if possible. $$\frac{2 y}{x^{2}-y^{2}}+\frac{2 x}{y^{2}-x^{2}}$$

Simplify each rational expression. If the rational expression cannot be simplified, so state. $$\frac{5 x-4}{4-5 x}$$

Determine whether statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. \(\frac{\frac{1}{4}-\frac{1}{3}}{\frac{1}{3}+\frac{1}{6}}=\frac{1}{12} \div \frac{3}{6}=\frac{1}{6}\)

Add or subtract as indicated. Simplify the result, if possible. $$\frac{2 x-1}{x+6}-\frac{6-5 x}{x^{2}-36}$$

Divide as indicated. $$\frac{4 x^{2}-y^{2}}{x^{2}+4 x y+4 y^{2}}+\frac{4 x-2 y}{3 x+6 y}$$

denominators are opposites, or additive inverses. Add or subtract as indicated. Simplify the result, if possible. $$\frac{x^{2}-2}{x^{2}+6 x-7}+\frac{19-4 x}{7-6 x-x^{2}}$$

Simplify each rational expression. If the rational expression cannot be simplified, so state. $$\frac{x-5}{x+5}$$

Add or subtract as indicated. Simplify the result, if possible. $$\frac{y^{2}-39}{y^{2}+3 y-10}-\frac{y-7}{y-2}$$

Divide as indicated. $$\frac{x y-y^{2}}{x^{2}+2 x+1} \div \frac{2 x^{2}+x y-3 y^{2}}{2 x^{2}+5 x y+3 y^{2}}$$

What is a rational equation?

denominators are opposites, or additive inverses. Add or subtract as indicated. Simplify the result, if possible. $$\frac{2 x+3}{x^{2}-x-30}+\frac{x-2}{30+x-x^{2}}$$

Simplify each rational expression. If the rational expression cannot be simplified, so state. $$\frac{x-7}{x+7}$$

Add or subtract as indicated. Simplify the result, if possible. $$\frac{y^{2}-6}{y^{2}+9 y+18}-\frac{y-4}{y+6}$$

Divide as indicated. $$\frac{x^{2}-4 y^{2}}{x^{2}+3 x y+2 y^{2}} \div \frac{x^{2}-4 x y+4 y^{2}}{x+y}$$

Explain how to solve a rational equation.

perform the indicated operation or operations. Simplify the result, if possible. $$\frac{6 b^{2}-10 b}{16 b^{2}-48 b+27}+\frac{7 b^{2}-20 b}{16 b^{2}-48 b+27}-\frac{6 b-3 b^{2}}{16 b^{2}-48 b+27}$$

Simplify each rational expression. If the rational expression cannot be simplified, so state. $$\frac{4 x-6}{3-2 x}$$

Simplify completely. \(\frac{2 y}{2+\frac{2}{y}}+\frac{y}{1+\frac{1}{y}}\)

Add or subtract as indicated. Simplify the result, if possible. $$4+\frac{1}{x-3}$$

Perform the indicated operation or operations. $$\left(\frac{y-2}{y^{2}-9 y+18} \cdot \frac{y^{2}-4 y-12}{y+2}\right) \div \frac{y^{2}-4}{y^{2}+5 y+6}$$

Explain how to find restrictions on the variable in a rational equation.

perform the indicated operation or operations. Simplify the result, if possible. $$\frac{22 b+15}{12 b^{2}+52 b-9}+\frac{30 b-20}{12 b^{2}+52 b-9}-\frac{4-2 b}{12 b^{2}+52 b-9}$$

Simplify each rational expression. If the rational expression cannot be simplified, so state. $$\frac{9 x-15}{5-3 x}$$

Simplify completely. \(\frac{1+\frac{1}{y}-\frac{6}{y^{2}}}{1-\frac{5}{y}+\frac{6}{y^{2}}}-\frac{1-\frac{1}{y}}{1-\frac{2}{y}-\frac{3}{y^{2}}}\)

Add or subtract as indicated. Simplify the result, if possible. $$7+\frac{1}{x-5}$$

Perform the indicated operation or operations. $$\left(\frac{6 y^{2}+31 y+18}{3 y^{2}-20 y+12} \cdot \frac{2 y^{2}-15 y+18}{6 y^{2}+35 y+36}\right) \div \frac{2 y^{2}-13 y+15}{9 y^{2}+15 y+4}$$

Why should restrictions on the variable in a rational equation be listed before you begin solving the equation?

perform the indicated operation or operations. Simplify the result, if possible. $$\frac{2 y}{y-5}-\left(\frac{2}{y-5}+\frac{y-2}{y-5}\right)$$

Simplify each rational expression. If the rational expression cannot be simplified, so state. $$\frac{4-6 x}{3 x^{2}-2 x}$$

Use the \([\text { GRAPH }]\) or \([\text { TABLE }]\) feature of a graphing utility to determine if the simplification is correct. If the answer is wrong, correct it and then verify your corrected simplification using the graphing utility. \(\frac{x-\frac{1}{2 x+1}}{1-\frac{x}{2 x+1}}=2 x-1\)

Add or subtract as indicated. Simplify the result, if possible. $$3-\frac{3 y}{y+1}$$

Perform the indicated operation or operations. $$\frac{3 x^{2}+3 x-60}{2 x-8}+\left(\frac{30 x^{2}}{x^{2}-7 x+10} \cdot \frac{x^{3}+3 x^{2}-10 x}{25 x^{3}}\right)$$

Describe similarities and differences between the procedures needed to solve the following problems: $$ \text { Add: } \frac{2}{x}+\frac{3}{4}, \quad \text { Solve: } \frac{2}{x}+\frac{3}{4}=1 $$