Chapter 9

Find any points of discontinuity for each rational function. $$ y=\frac{x^{2}}{x^{2}+1} $$

Find the least common multiple of each pair of polynomials. \(5 y^{2}-80\) and \(y+4\)

Draw a graph of each function. Describe properties of the graph. \(y=\frac{100}{x}\)

Is the relationship between the values in each table a direct variation, an inverse variation, or neither? Write equations to model the direct and inverse variations. $$ \begin{array}{|c|c|c|c|c|}\hline x & {3} & {5} & {7} & {10.5} \\ \hline y & {14} & {8.4} & {6} & {4} \\ \hline\end{array} $$

Suppose you have five books in your book bag. Three are novels, one is a biography, and one is a poetry book. Today you grab one book out of your bag without looking, and return it later. Tomorrow you do the same thing. What is the probability that you grab a novel both days?

Solve each equation. Check each solution. $$ \frac{2}{3 x-5}=\frac{4}{x-15} $$

Multiply. State any restrictions on the variables. $$ \frac{8 y-4}{10 y-5} \cdot \frac{5 y-15}{3 y-9} $$

Find any points of discontinuity for each rational function. $$ y=\frac{1}{2 x^{2}+3 x-7} $$

Find the least common multiple of each pair of polynomials. \(x^{2}-32 x-10\) and \(2 x+10\)

Draw a graph of each function. Describe properties of the graph. \(y=\frac{-0.1}{x}\)

Is the relationship between the values in each table a direct variation, an inverse variation, or neither? Write equations to model the direct and inverse variations. $$ \begin{array}{|c|c|c|c|c|}\hline x & {0.5} & {2.1} & {3.5} & {11} \\ \hline y & {1} & {4.2} & {7} & {22} \\ \hline\end{array} $$

Two standard number cubes are tossed. State whether the events are mutually exclusive. Explain your reasoning. The sum is a prime number; the sum is less than 4

Solve each equation. Check each solution. $$ \frac{1}{4}-x=\frac{x}{8} $$

Multiply. State any restrictions on the variables. $$ \frac{2 x+12}{3 x-9} \cdot \frac{2 x-6}{3 x+8} $$

Describe the vertical asymptotes and holes for the graph of each rational function. $$ y=\frac{3}{x+2} $$

Simplify each sum. \(\frac{1}{2 x}+\frac{1}{2 x}\)

The weight \(P\) in pounds that a beam can safely carry is inversely proportional to the distance \(D\) in feet between the supports of the beam. For a certain type of wooden beam, \(P=\frac{9200}{D} .\) Use a graphing calculator and the Intersect feature to find the distance between supports that is needed to carry each given weight. 500 \(\mathrm{lb}\)

Is the relationship between the values in each table a direct variation, an inverse variation, or neither? Write equations to model the direct and inverse variations. $$ \begin{array}{|c|c|c|c|c|}\hline x & {0.1} & {3} & {6} & {24} \\ \hline y & {3} & {0.1} & {0.05} & {0.0125} \\ \hline\end{array} $$

Two standard number cubes are tossed. State whether the events are mutually exclusive. Explain your reasoning. The numbers are equal; the sum is odd.

Solve each equation. Check each solution. $$ \frac{y}{5}+\frac{y}{2}=7 $$

Multiply. State any restrictions on the variables. $$ \frac{x^{2}-4}{x^{2}-1} \cdot \frac{x+1}{x^{2}+2 x} $$

Describe the vertical asymptotes and holes for the graph of each rational function. $$ y=\frac{x+5}{x+5} $$

Simplify each sum. \(\frac{d-3}{2 d+1}+\frac{d-1}{2 d+1}\)

The weight \(P\) in pounds that a beam can safely carry is inversely proportional to the distance \(D\) in feet between the supports of the beam. For a certain type of wooden beam, \(P=\frac{9200}{D} .\) Use a graphing calculator and the Intersect feature to find the distance between supports that is needed to carry each given weight. 1200 \(\mathrm{lb}\)

Is the relationship between the values in each table a direct variation, an inverse variation, or neither? Write equations to model the direct and inverse variations. $$ \begin{array}{|c|c|c|c|c|}\hline x & {7} & {3} & {1} & {\frac{1}{5}} \\\ \hline y & {\frac{1}{7}} & {\frac{1}{3}} & {1} & {5} \\ \hline\end{array} $$

Two standard number cubes are tossed. State whether the events are mutually exclusive. Explain your reasoning. The product is greater than \(20 ;\) the product is a multiple of \(3 .\)

Solve each equation. Check each solution. $$ \frac{2 x}{3}-\frac{1}{2}=\frac{2 x+5}{6} $$

Multiply. State any restrictions on the variables. $$ \frac{x^{2}-5 x+6}{x^{2}-4} \cdot \frac{x^{2}+3 x+2}{x^{2}-2 x-3} $$

Describe the vertical asymptotes and holes for the graph of each rational function. $$ y=\frac{x+3}{(2 x+3)(x-1)} $$

Simplify each sum. \(\frac{5 y+2}{x y^{2}}+\frac{2 x-4}{4 x y}\)

The weight \(P\) in pounds that a beam can safely carry is inversely proportional to the distance \(D\) in feet between the supports of the beam. For a certain type of wooden beam, \(P=\frac{9200}{D} .\) Use a graphing calculator and the Intersect feature to find the distance between supports that is needed to carry each given weight. 2400 \(\mathrm{lb}\)

Is the relationship between the values in each table a direct variation, an inverse variation, or neither? Write equations to model the direct and inverse variations. $$ \begin{array}{|c|c|c|c|c|}\hline x & {10} & {12} & {20} & {23} \\ \hline y & {2} & {2 \frac{2}{5}} & {4} & {5 \frac{3}{5}} \\ \hline\end{array} $$

Two standard number cubes are tossed. State whether the events are mutually exclusive. Explain your reasoning. Population About 30\(\%\) of the U.S. population is under 20 years old. About 17\(\%\) of the population is over \(60 .\) What is the probability that a person chosen at random is under 20 or over 60\(?\)

Solve each equation. Check each solution. $$ \frac{3 x-2}{12}-\frac{1}{6}=\frac{1}{6} $$

Divide. State any restrictions on the variables. $$ \frac{7 x}{4 y^{3}} \div \frac{21 x^{3}}{8 y} $$

Describe the vertical asymptotes and holes for the graph of each rational function. $$ y=\frac{(x+3)(x-2)}{(x-2)(x+1)} $$

Simplify each sum. \(\frac{5 x}{x^{2}-9}+\frac{2}{x+4}\)

The weight \(P\) in pounds that a beam can safely carry is inversely proportional to the distance \(D\) in feet between the supports of the beam. For a certain type of wooden beam, \(P=\frac{9200}{D} .\) Use a graphing calculator and the Intersect feature to find the distance between supports that is needed to carry each given weight. 5000 \(\mathrm{lb}\)

Suppose that \(x\) and \(y\) vary inversely. Write a function that models each inverse variation and find \(y\) when \(x=10 .\) $$ x=20 \text { when } y=5 $$

\(\boldsymbol{S}\) and \(\boldsymbol{T}\) are mutually exclusive events. Find \(\boldsymbol{P}(\boldsymbol{S} \text { or } \boldsymbol{T})\) $$ P(S)=\frac{5}{8}, P(T)=\frac{1}{8} $$

Solve each equation. Check each solution. $$ \frac{1}{x}+\frac{x}{2}=\frac{x+4}{2 x} $$

Divide. State any restrictions on the variables. $$ \frac{3 x^{3}}{5 y^{2}} \div \frac{6 x^{5}}{5 y^{3}} $$

Describe the vertical asymptotes and holes for the graph of each rational function. $$ y=\frac{x^{2}-4}{x+2} $$

Simplify each sum. \(\frac{-3 x}{x^{2}-9}+\frac{4}{2 x-6}\)

Sketch the asymptotes and the graph of each equation. \(y=\frac{1}{x}-3\)

Suppose that \(x\) and \(y\) vary inversely. Write a function that models each inverse variation and find \(y\) when \(x=10 .\) $$ x=20 \text { when } y=-4 $$

\(\boldsymbol{S}\) and \(\boldsymbol{T}\) are mutually exclusive events. Find \(\boldsymbol{P}(\boldsymbol{S} \text { or } \boldsymbol{T})\) $$ P(S)=\frac{3}{5}, P(T)=\frac{1}{3} $$

Solve each equation. Check each solution. $$ \frac{11}{3 x}-\frac{1}{3}=\frac{-4}{x^{2}} $$

Divide. State any restrictions on the variables. $$ \frac{6 x+6 y}{x-y} \div \frac{18}{5 x-5 y} $$

Describe the vertical asymptotes and holes for the graph of each rational function. $$ y=\frac{x+5}{x^{2}+9} $$