Chapter 2

Solving an Equation Involving an Absolute Value Find all solutions of the equation algebraically. Check your solutions. $$|3 x+2|=7$$

Determine the intervals on which the polynomial is entirely negative and those on which it is entirely positive. $$-2 x^{2}+x+5$$

Use the Quadratic Formula to solve the equation. Use a graphing utility to verify your solutions graphically. $$x^{2}+16=-5 x$$

Use a graphing utility to approximate any solutions of the equation. [Remember to write the equation in the form \(f(x)=0.1\) $$8-\sqrt{x+9}=6$$

Write the quotient in standard form. $$\frac{8-7 i}{1-2 i}$$

Solving an Equation Involving an Absolute Value Find all solutions of the equation algebraically. Check your solutions. $$|x|=x^{2}+x-24$$

Determine the intervals on which the polynomial is entirely negative and those on which it is entirely positive. $$-x^{2}+6 x-10$$

Use the Quadratic Formula to solve the equation. Use a graphing utility to verify your solutions graphically. $$4 x=8-x^{2}$$

(a) Use a graphing utility to complete the table. Determine the interval in which the solution to the equation \(3.2 x-5.8=0\) is located. Explain your reasoning. $$\begin{array}{|l|l|l|l|l|l|l|} \hline x & -1 & 0 & 1 & 2 & 3 & 4 \\ \hline 3.2 x-5.8 & & & & & & \\ \hline \end{array}$$ (b) Use the graphing utility to complete the table. Determine the interval in which the solution to the equation \(3.2 x-5.8=0\) is located. Explain how this process can be used to approximate the solution to any desired degree of accuracy. Then use the graphing utility to verify graphically the solution to \(3.2 x-5.8=0\) $$\begin{array}{|l|l|l|l|l|l|l|} \hline x & 1.5 & 1.6 & 1.7 & 1.8 & 1.9 & 2 \\ \hline 3.2 x-5.8 & & & & & & \\ \hline \end{array}$$

A certificate of deposit with an initial deposit of \(\$ 8000\) accumulates \(\$ 200\) interest in 2 years. Find the annual simple interest rate.

Write the quotient in standard form. $$\frac{i}{(4-5 i)^{2}}$$

Solving an Equation Involving an Absolute Value Find all solutions of the equation algebraically. Check your solutions. $$\left|x^{2}+6 x\right|=3 x+18$$

Determine the intervals on which the polynomial is entirely negative and those on which it is entirely positive. $$3 x^{2}+8 x+6$$

Use the Quadratic Formula to solve the equation. Use a graphing utility to verify your solutions graphically. $$8 x=4-x^{2}$$

You plan to invest \(\$ 12,000\) in two funds paying \(4 \frac{1}{2} \%\) and \(5 \%\) simple interest. (There is more risk in the \(5 \%\) fund.) Your goal is to obtain a total annual interest income of \(\$ 560\) from the investments. What is the least amount you can invest in the \(5 \%\) fund to meet your objective?

Write the quotient in standard form. $$\frac{5 i}{(2+3 i)^{2}}$$

Solving an Equation Involving an Absolute Value Find all solutions of the equation algebraically. Check your solutions. $$|x+1|=x^{2}-5$$

Solve the inequality and graph the solution on the real number line. Use a graphing utility to verify your solution graphically. $$x^{2}+4 x+4 \geq 9$$

Use the Quadratic Formula to solve the equation. Use a graphing utility to verify your solutions graphically. $$20 x^{2}-20 x+5=0$$

Determine any point(s) of intersection algebraically. Then verify your result numerically by creating a table of values for each function. $$\begin{aligned} &y=6-x\\\ &y=3 x-2 \end{aligned}$$

A grocer mixes peanuts that cost \(\$ 2.50\) per pound and walnuts that cost \(\$ 8.00\) per pound to make 100 pounds of a mixture that costs \(\$ 5.25\) per pound. How much of each kind of nut is put into the mixture?

Perform the operation and write the result in standard form. $$\frac{2}{1+i}-\frac{3}{1-i}$$

Solving an Equation Involving an Absolute Value Find all solutions of the equation algebraically. Check your solutions. $$|x-15|=x^{2}-15 x$$

Solve the inequality and graph the solution on the real number line. Use a graphing utility to verify your solution graphically. $$x^{2}-6 x+9<16$$

Use the Quadratic Formula to solve the equation. Use a graphing utility to verify your solutions graphically. $$9 x^{2}-18 x+9=0$$

Determine any point(s) of intersection algebraically. Then verify your result numerically by creating a table of values for each function. $$\begin{aligned} &y=2 x-3\\\ &y=9-x \end{aligned}$$

A forester mixes gasoline and oil to make 2 gallons of mixture for a two-cycle chainsaw engine. This mixture is 32 parts gasoline and 1 part oil. How much gasoline must be added to bring the mixture to 40 parts gasoline and 1 part oil?

Perform the operation and write the result in standard form. $$\frac{2 i}{2+i}+\frac{5}{2-i}$$

Graphical Analysis (a) use a graphing utility to graph the equation, (b) use the graph to approximate any \(x\) -intercepts of the graph, (c) set \(y=0\) and solve the resulting equation, and (d) compare the result of part (c) with the \(x\) -intercepts of the graph. $$y=x^{3}-2 x^{2}-3 x$$

Solve the inequality and graph the solution on the real number line. Use a graphing utility to verify your solution graphically. $$(x+2)^{2}<25$$

Use the Quadratic Formula to solve the equation. Use a graphing utility to verify your solutions graphically. $$16 x^{2}+24 x+9=0$$

Determine any point(s) of intersection algebraically. Then verify your result numerically by creating a table of values for each function. $$\begin{aligned} &2 x+y=6\\\ &-x+y=0 \end{aligned}$$

A store has \(\$ 40,000\) of inventory in notebook computers and tablet computers. The profit on a notebook computer is \(20 \%\) and the profit on a tablet computer is \(25 \%\). The profit for the entire stock is \(24 \% .\) How much is invested in notebook computers and how much in tablet computers?

Perform the operation and write the result in standard form. $$\frac{i}{3-2 i}+\frac{2 i}{3+8 i}$$

Graphical Analysis (a) use a graphing utility to graph the equation, (b) use the graph to approximate any \(x\) -intercepts of the graph, (c) set \(y=0\) and solve the resulting equation, and (d) compare the result of part (c) with the \(x\) -intercepts of the graph. $$y=2 x^{4}-15 x^{3}+18 x^{2}$$

Solve the inequality and graph the solution on the real number line. Use a graphing utility to verify your solution graphically. $$(x-3)^{2} \geq 1$$

Use the Quadratic Formula to solve the equation. Use a graphing utility to verify your solutions graphically. $$9 x^{2}+30 x+25=0$$

Determine any point(s) of intersection algebraically. Then verify your result numerically by creating a table of values for each function. $$\begin{aligned} &x-y=-4\\\ &x+2 y=5 \end{aligned}$$

A store has \(\$ 4500\) of inventory in \(8 \times 10\) picture frames and \(5 \times 7\) picture frames. The profit on an \(8 \times 10\) frame is \(25 \%\) and the profit on a \(5 \times 7\) frame is \(22 \%\). The profit on the entire stock is \(24 \% .\) How much is invested in the \(8 \times 10\) picture frames and how much in the \(5 \times 7\) picture frames?

Perform the operation and write the result in standard form. $$\frac{1+i}{i}-\frac{3}{4-i}$$

Graphical Analysis (a) use a graphing utility to graph the equation, (b) use the graph to approximate any \(x\) -intercepts of the graph, (c) set \(y=0\) and solve the resulting equation, and (d) compare the result of part (c) with the \(x\) -intercepts of the graph. $$y=x^{4}-10 x^{2}+9$$

Solve the inequality and graph the solution on the real number line. Use a graphing utility to verify your solution graphically. $$x^{3}-4 x^{2} \geq 0$$

Use the Quadratic Formula to solve the equation. Use a graphing utility to verify your solutions graphically. $$4 x^{2}+16 x+17=0$$

Determine any point(s) of intersection algebraically. Then verify your result numerically by creating a table of values for each function. $$\begin{aligned} &x-y=10\\\ &x+2 y=4 \end{aligned}$$

A triangular sail has an area of 182.25 square feet. The sail has a base of 13.5 feet. Find the height of the sail.

Simplify the complex number and write it in standard form. $$-6 i^{3}+i^{2}$$

Graphical Analysis (a) use a graphing utility to graph the equation, (b) use the graph to approximate any \(x\) -intercepts of the graph, (c) set \(y=0\) and solve the resulting equation, and (d) compare the result of part (c) with the \(x\) -intercepts of the graph. $$y=x^{4}-29 x^{2}+100$$

Solve the inequality and graph the solution on the real number line. Use a graphing utility to verify your solution graphically. $$x^{5}-3 x^{4} \leq 0$$

Use the Quadratic Formula to solve the equation. Use a graphing utility to verify your solutions graphically. $$9 x^{2}-6 x+37=0$$

Determine any point(s) of intersection algebraically. Then verify your result numerically by creating a table of values for each function. $$\begin{aligned} &4 x-y=4\\\ &x-4 y=1 \end{aligned}$$