Chapter 8

HEALTH The prediction equation \(y=205-0.5 x\) relates a person's maximum heart rate for exercise \(y\) and age \(x\) . Use the equation to find the maximum heart rate for an 18 -year-old.

Find the LCM of each set of polynomials. \(2 t^{2}-9 t-5, t^{2}+t-30\)

For Exercises \(53-55,\) use the following information. Jalisa is competing in a 48 -mile bicycle race. She travels half the distance at one rate. The rest of the distance, she travels 4 miles per hour slower. If \(x\) represents the faster pace in miles per hour, write an expression that represents the time spent at that pace.

CHALLENGE Write three rational functions that have a vertical asymptote at \(x=3\) and a hole at \(x=-2 .\)

Determine the equations of any vertical asymptotes and the values of x for any holes in the graph of each rational function. \(f(x)=\frac{x+1}{x^{2}-1}\)

Under what conditions is \(\frac{a^{2}+a b+b^{2}}{a^{2}-b^{2}}\) undefined?

Determine the value of \(r\) so that a line through the points with the given coordinates has the given slope. (Lesson \(2-3 )\) $$ (r, 2),(4,-6) ; \text { slope }=-\frac{8}{3} $$

Determine the equations of any vertical asymptotes and the values of x for any holes in the graph of each rational function. \(f(x)=\frac{x+3}{x^{2}+x-12}\)

At the end of the 2005-2006 season, the Seattle Sonics’ Ray Allen had made 5422 field goals out of 12,138 attempts during his NBA career. Write a ratio to represent the ratio of the number of career field goals made to career field goals attempted by Ray Allen at the end of the 2005-2006 season.

Determine the value of \(r\) so that a line through the points with the given coordinates has the given slope. (Lesson \(2-3 )\) $$ (r, 6),(8,4) ; \text { slope }=\frac{1}{2} $$

Determine the equations of any vertical asymptotes and the values of x for any holes in the graph of each rational function. \(f(x)=\frac{x^{2}+4 x+3}{x+3}\)

At the end of the 2005-2006 season, the Seattle Sonics’ Ray Allen had made 5422 field goals out of 12,138 attempts during his NBA career. Suppose Ray Allen attempted \(a\) field goals and made \(m\) field goals during the 2006-2007 season. Write a rational expression to represent the ratio of the number of career field goals made to the number of career field goals attempted at the end of the 2006-2007 season.

MAGNETS For a bar magnet, the magnetic field strength \(H\) at a point \(P\) along the axis of the magnet is \(H=\frac{m}{2 L(d-L)^{2}}-\frac{m}{2 L(d+L)^{2}}\) . Write a simpler expression for \(H .\)

REVIEW \(\frac{x+2}{x+3}+\frac{4}{x^{2}+x-6}=\) $$ \begin{array}{ll}{\mathbf{F} \frac{-3 x-9}{x^{2}+x-6}} & {\mathbf{H} \frac{x^{2}}{x^{2}+x-6}} \\ {\mathbf{G} \frac{x^{2}-3 x-24}{x^{2}+x-6}} & {\mathbf{J} \frac{x^{2}+x-1}{x^{2}+x-6}}\end{array} $$

Simplify each expression. \(\frac{3 x}{x-y}+\frac{4 x}{y-x}\)

An airplane is traveling at the rate \(r\) of 500 miles per hour for a time \(t\) of \((6+x)\) hours. A second airplane travels at the rate \(r\) of \((540+90 x)\) miles per hour for a time \(t\) of 6 hours. Write a rational expression to represent the ratio of the distance \(d\) traveled by the first airplane to the distance \(d\) traveled by the second airplane.

Write two polynomials that have a LCM of \(d^{3}-d\)

Simplify each expression. $$ \frac{3 m+2}{m+n}+\frac{4}{2 m+2 n} $$

Simplify each expression. \(\frac{t}{t+2}-\frac{2}{t^{2}-4}\)

An airplane is traveling at the rate \(r\) of 500 miles per hour for a time \(t\) of \((6+x)\) hours. A second airplane travels at the rate \(r\) of \((540+90 x)\) miles per hour for a time \(t\) of 6 hours. Simplify the rational expression. What does this expression tell you about the distances traveled of the two airplanes?

Simplify each expression. $$ \frac{5}{x+3}-\frac{2}{x-2} $$

Simplify each expression. \(\frac{m-\frac{1}{m}}{1+\frac{4}{m}-\frac{5}{m^{2}}}\)

Find two rational expressions whose sum is \(\frac{2 x-1}{(x+1)(x-2)}\)

Simplify each expression. $$ \frac{2 w-4}{w+3} \div \frac{2 w+6}{5} $$

One estimate for the number of cells in the human body is 100,000,000,000,000. Write this number in scientific notation.

Consider \(f(x)=\frac{-15 x^{2}+10 x}{5 x}\) and \(g(x)=-3 x+2\). Simplify \(\frac{-15 x^{2}+10 x}{5 x} .\) What do you observe about the expression?

REASONING In the expression \(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}, a, b,\) and \(c\) are nonzero real numbers. Determine whether each statement is sometimes, always, or never true. Explain your answer. a. \(a b c\) is a common denominator. b. \(a b c\) is the LCD. c. \(a b\) is the LCD. d. \(b\) is the LCD. e. The sum is \(\frac{b c+a c+a b}{a b c}\)

Find all of the rational zeros for each function. $$ f(x)=x^{3}+5 x^{2}+2 x-8 $$

State the slope and the y-intercept of the graph of each equation. \(y=0.4 x+1.2\)

Consider \(f(x)=\frac{-15 x^{2}+10 x}{5 x}\) and \(g(x)=-3 x+2\). Graph \(f(x)\) and \(g(x)\) on a graphing calculator. How do the graphs appear?

Find all of the rational zeros for each function. $$ g(x)=2 x^{3}-9 x^{2}+7 x+6 $$

State the slope and the y-intercept of the graph of each equation. \(2 y=6 x+14\)

Consider \(f(x)=\frac{-15 x^{2}+10 x}{5 x}\) and \(g(x)=-3 x+2\). Use the table feature to examine the function values for \(f(x)\) and \(g(x) .\) How do the tables compare?

ACT/SAT What is the sum of \(\frac{x-y}{5}\) and \(\frac{x+y}{4} ?\) $$ \begin{array}{l}{\text { A } \frac{x+9 y}{20}} \\ {\text { B } \frac{9 x+y}{20}} \\ {\text { C } \frac{9 x-y}{20}} \\ {\text { D } \frac{x-9 y}{20}}\end{array} $$

ART Joyce Jackson purchases works of art for an art gallery. Two years ago she bought a painting for \(\$ 20,000,\) and last year she bought one for \(\$ 35,000\) . If paintings appreciate 14\(\%\) per year, how much are the two paintings worth now? (Lesson \(6-5 )\)

State the slope and the y-intercept of the graph of each equation. \(3 x+5 y=15\)

Consider \(f(x)=\frac{-15 x^{2}+10 x}{5 x}\) and \(g(x)=-3 x+2\). How can you use what you have observed with \(f(x)\) and \(g(x)\) to verify that expressions are equivalent or to identify excluded values?

Given: Two angles are complementary The measure of one angle is \(15^{\circ}\) more than the measure of the other angle. Conclusion: The measures of the angles are \(30^{\circ}\) and \(45^{\circ} .\) This conclusion - F is contradicted by the first statement given. G is verified by the first statement given. H invalidates itself because a \(45^{\circ}\) angle cannot be complementary to another. J verifies itself because \(30^{\circ}\) is \(15^{\circ}\) less than \(45^{\circ}\) .

Solve each equation by completing the square. $$ x^{2}+8 x+20=0 $$

Identify each function as S for step, C for constant, A for absolute value, or P for piecewise. \(h(x)=\frac{2}{3}\)

Write two rational expressions that are equivalent.

Simplify each expression. $$ \frac{9 x^{2} y^{3}}{(5 x y z)^{2}} \div \frac{(3 x y)^{3}}{20 x^{2} y} $$

Solve each equation by completing the square. $$ x^{2}+2 x-120=0 $$

Identify each function as S for step, C for constant, A for absolute value, or P for piecewise. \(g(x)=3|x|\)

Rewrite \(\frac{a+\sqrt{b}}{-a^{2}+b}\) so it has a numerator of 1.

Simplify each expression. $$ \frac{5 a^{2}-20}{2 a+2} \cdot \frac{4 a}{10 a-20} $$

Solve each equation by completing the square. $$ x^{2}+7 x-17=0 $$

Graph \(y \leq \sqrt{x+1}\)

Write the slope-intercept form of the equation for the line that passes through \((1,-2)\) and is perpendicular to the line with equation \(y=-\frac{1}{5} x+2 .(\operatorname{losson} 2-4)\)

Identify each function as S for step, C for constant, A for absolute value, or P for piecewise. \(f(x)=\left\\{\begin{array}{c}{1 \text { if } x>0} \\ {-1 \text { if } x \leq 0}\end{array}\right.\)