Chapter 6

Solve each equation. $$ |2 x-1|=9 $$

Perform the operations and simplify the result when possible. $$x+1+\frac{1}{x-1}$$

Use similar triangles to solve each problem. Ski Runs. A ski course with \(\frac{1}{2}\) mile of horizontal run falls 100 feet in every 300 feet of run. Find the height of the hill.

Perform the operations and simplify. $$ \frac{2 p^{2}-5 p q-3 q^{2}}{p^{2}-9 q^{2}} \div \frac{2 p^{2}+5 p q+2 q^{2}}{2 p^{2}+5 p q-3 q^{2}} $$

Simplify each expression. If an expression cannot be simplified, write "Does not simplify." $$ \frac{y-x y}{x y-x} $$

Use synthetic division to perform each division. $$ \left(2 x^{3}-50-16 x^{2}-35 x\right) \div(x-10) $$

Solve equation. If a solution is extraneous, so indicate. \(3 x^{-2}-4 x^{-1}+1=0\) \(\left(\text {Hint: Use } x^{-n}=\frac{1}{x^{n}}\right)\)

Simplify each expression. $$ a+\frac{a}{1+\frac{a}{a+1}} $$

Perform the operations and simplify the result when possible. $$\frac{d}{d^{2}+11 d+30}-\frac{5}{d^{2}+9 d+20}$$

The language of variation is often used to describe various aspects of the Internet and websites. Determine whether each statement, generally speaking, is true or false. a. The dollar amount of sales that an Internet website receives is inversely proportional to the amount of Internet traffic that visits the website. b. The download time of an Internet website varies directly with the bandwidth being used. c. Search engines like Google place a value on a website that is directly proportional to the number of sites that link to it.

Let \(f(x)=4 x^{4}+20 x^{3}-x^{2}-2 x+15\) and \(g(x)=x+5\) Find \(\frac{f(x)}{g(x)}\) in simplified form.

Perform the operations and simplify. $$ \left(4 x^{2}-9\right) \div \frac{2 x^{2}+5 x+3}{x+2} \cdot \frac{1}{2 x-3} $$

Simplify each expression. If an expression cannot be simplified, write "Does not simplify." $$ \frac{x^{4}+3 x^{3}+9 x^{2}}{x^{3}-27} $$

Use synthetic division to perform each division. $$ \left(m^{3}-m^{2}-m-1\right) \div(m-1) $$

Solve equation. If a solution is extraneous, so indicate. \(3 y^{-2}-y^{-1}-2=0\) \(\left(\text {Hint: Use } x^{-n}=\frac{1}{x^{n}}\right)\)

Simplify each expression. $$ b+\frac{b}{1-\frac{b+1}{b}} $$

Perform the operations and simplify the result when possible. $$\frac{t}{t^{2}+9 t+20}-\frac{4}{t^{2}+7 t+12}$$

Perform each division. Let \(s(t)=t^{5}-t^{4}+7 t^{2}-27 t+10\) and \(h(t)=t^{2}-t+5\) Find \(\frac{s(t)}{h(t)}\) in simplified form.

Perform the operations and simplify. $$ (4 x+12) \div \frac{2 x-6}{x^{2}} \cdot \frac{x-3}{2} $$

Simplify each expression. If an expression cannot be simplified, write "Does not simplify." $$ \frac{x^{3}+8}{x^{4}-2 x^{3}+4 x^{2}} $$

Use synthetic division to perform each division. $$ \left(4 x^{3}-1+5 x^{2}\right) \div(x+2) $$

Solve equation. If a solution is extraneous, so indicate. \(\frac{5}{2 z^{2}+z-3}-\frac{2}{2 z+3}=\frac{z+1}{z-1}-1\)

Simplify each expression. $$ \frac{x-\frac{1}{1-\frac{x}{2}}}{\frac{3}{x+\frac{2}{3}}+x} $$

Perform the operations and simplify the result when possible. $$\frac{3}{x+1}-\frac{2}{x-1}+\frac{x+3}{x^{2}-1}$$

Solve each problem by writing a variation model. Gravity. The force of gravity acting on an object varies directly as the mass of the object. The force on a mass of 5 kilograms is 49 newtons. What is the force acting on a mass of 12 kilograms?

Perform each division. \(\frac{4 x^{3}+4 x^{2}+7 x-5}{x-\frac{1}{2}}\)

Perform the operations and simplify. $$ \frac{x^{3}-3 x^{2}-25 x+75}{x^{3}-27} \cdot \frac{2 x^{3}+6 x^{2}+18 x}{x^{2}+10 x+25} $$

Simplify each expression. If an expression cannot be simplified, write "Does not simplify." $$ \frac{2 x^{2}+2 x-12}{x^{3}+3 x^{2}-4 x-12} $$

Use synthetic division to perform each division. $$ \left(t^{3}+t^{2}+t+2\right) \div(t+1) $$

Solve equation. If a solution is extraneous, so indicate. \(\frac{x}{x-5}+\frac{5}{x}=\frac{11}{6}\)

Simplify each expression. $$ \frac{3 x-\frac{1}{3-\frac{x}{2}}}{\frac{3}{\frac{x}{2}-3}+x} $$

Perform the operations and simplify the result when possible. $$\frac{7 n^{2}}{m-n}+\frac{3 m}{n-m}-\frac{3 m^{2}-n}{m^{2}-2 m n+n^{2}}$$

Solve each problem by writing a variation model. Free Fall. An object in free fall travels a distance \(s\) that is directly proportional to the square of the time \(t\). If an object falls \(1,024\) feet in 8 seconds, how far will it fall in 10 seconds?

Perform the operations and simplify. $$ \frac{x^{2}+3 x+x y+3 y}{x^{2}-9} \cdot \frac{3-x}{x^{3}+3 x^{2}} $$

Simplify each expression. If an expression cannot be simplified, write "Does not simplify." $$ \frac{3 x^{2}-3 y^{2}}{x^{2}+2 y+2 x+y x} $$

Use synthetic division to perform each division. Divide \(8 a^{3}-10 a^{2}-32 a-15\) by \(a+\frac{3}{4}\)

Solve equation. If a solution is extraneous, so indicate. \(\frac{5}{3 x+12}-\frac{1}{9}=\frac{x-1}{3 x}\)

Simplify each expression. $$ \left(x^{-1} y^{-1}\right)\left(x^{-1}+y^{-1}\right)^{-1} $$

Perform the operations and simplify the result when possible. $$\frac{8}{9 y^{2}}+\frac{1}{6 y^{4}}$$

Solve each problem by writing a variation model. Finding Distance. The distance that a car can go varies directly as the number of gallons of gasoline it consumes. If a car can go 288 miles on 12 gallons of gasoline, how far can it go on a full tank of 18 gallons?

a. \(\frac{16 n^{2}-16 n-5}{4 n}\) b. \(\frac{16 n^{2}-16 n-5}{4 n+1}\)

Perform the operations and simplify. $$ \begin{aligned} &\text { Let } f(x)=\frac{x^{2}+x-6}{x^{2}-6 x+9} \text { and } g(x)=\frac{x^{2}-9}{x^{2}-4}\\\ &\text { Find } f(x) \cdot g(x) \end{aligned} $$

Simplify each expression. If an expression cannot be simplified, write "Does not simplify." $$ \frac{4 x^{2}+8 x+3}{6+x-2 x^{2}} $$

Use synthetic division to perform each division. Divide \(4 a^{3}-2 a^{2}-18 a-9\) by \(a+\frac{3}{2}\)

Solve equation. If a solution is extraneous, so indicate. \(\frac{1}{y+5}=\frac{1}{3 y+6}-\frac{y+2}{y^{2}+7 y+10}\)

Simplify each expression. $$ \left[\left(x^{-1}+1\right)^{-1}+1\right]^{-1} $$

Perform the operations and simplify the result when possible. $$\frac{5}{6 a^{3}}+\frac{7}{8 a^{2}}$$

Solve each problem by writing a variation model. Braking. Suppose the distance that a vehicle travels after its brakes have been applied varies directly as the square of the speed at which it was traveling. If the stopping distance for such a vehicle going 20 mph is 24 feet, what is the stopping distance for the vehicle traveling at 50 mph?

a. \(\frac{9 a^{3}+3 a^{2}+4 a+4}{3 a}\) b. \(\frac{9 a^{3}+3 a^{2}+4 a+4}{3 a+2}\)

Perform the operations and simplify. $$ \begin{aligned} &\text { Let } g(s)=\frac{s^{2}-5 s+6}{s^{2}-10 s+16} \text { and } h(s)=\frac{s^{2}-6 s-16}{s^{2}+2 s}\\\ &\text { Find } g(s) \cdot h(s) \end{aligned} $$