Chapter 1

Solve equation by completing the square. $$ x^{2}+3 x-1=0 $$

Solve each equation in Exercises 41–60 by making an appropriate substitution. $$\left(x^{2}-x\right)^{2}-14\left(x^{2}-x\right)+24=0$$

In your own words, describe a step-by-step approach for solving algebraic word problems.

The mathematician Girolamo Cardano is credited with the first use (in 1545 ) of negative square roots in solving the now-famous problem, "Find two numbers whose sum is 10 and whose product is \(40 . "\) Show that the complex numbers \(5+i \sqrt{15}\) and \(5-i \sqrt{15}\) satisfy the conditions of the problem. (Cardano did not use the symbolism \(i \sqrt{15}\) or even \(\sqrt{-15} .\) He wrote \(\mathrm{R} \cdot \mathrm{m} 15\) for \(\sqrt{-15},\) meaning "radix minus \(15 .^{\prime \prime}\) He regarded the numbers \(5+\mathrm{R} . \mathrm{m} 15\) and \(5-\mathrm{R} . \mathrm{m} 15\) as "fictitious" or "ghost numbers," and considered the problem "manifestly impossible." But in a mathematically adventurous spirit, he exclaimed," Nevertheless, we will operate.")

Find all values of \(x\) such that \(y=0\) $$ y=4[x-(3-x)]-7(x+1) $$

In Exercises 51–58, solve each compound inequality. $$ -6 \leq \frac{1}{2} x-4<-3 $$

Solve equation by completing the square. $$ x^{2}-3 x-5=0 $$

Solve each equation in Exercises 41–60 by making an appropriate substitution. $$\left(x^{2}-2 x\right)^{2}-11\left(x^{2}-2 x\right)+24=0$$

Write an original word problem that can be solved using a linear equation. Then solve the problem.

Find all values of \(x\) such that \(y=0\) $$y=2[3 x-(4 x-6)]-5(x-6)$$

In Exercises 59–94, solve each absolute value inequality. $$ |x|<3 $$

Solve equation by completing the square. $$ 2 x^{2}-7 x+3=0 $$

Solve each equation in Exercises 41–60 by making an appropriate substitution. $$\left(y-\frac{8}{y}\right)^{2}+5\left(y-\frac{8}{y}\right)-14=0$$

Find all values of \(x\) such that \(y=0\) $$y=\frac{x+6}{3 x-12}-\frac{5}{x-4}-\frac{2}{3}$$

Explain how to add complex numbers. Provide an example with your explanation.

In Exercises 59–94, solve each absolute value inequality. $$ |x|<5 $$

Solve equation by completing the square. $$ 2 x^{2}+5 x-3=0 $$

Solve each equation in Exercises 41–60 by making an appropriate substitution. $$\left(y-\frac{10}{y}\right)^{2}+6\left(y-\frac{10}{y}\right)-27=0$$

Did you have difficulties solving some of the problems that were assigned in this Exercise Set? Discuss what you did if this happened to you. Did your course of action enhance your ability to solve algebraic word problems?

Explain how to multiply complex numbers and give an example.

In Exercises 59–94, solve each absolute value inequality. $$ |x-1| \leq 2 $$

Solve equation by completing the square. $$ 4 x^{2}-4 x-1=0 $$

In Exercises 61–78, solve each absolute value equation or indicate that the equation has no solution. $$|x|=8$$

Explaining the Concepts What is the rectangular coordinate system?

Solve each equation. Then determine whether the equation is an identity, a conditional equation, or an inconsistent equation. $$5 x+9=9(x+1)-4 x$$

What is the complex conjugate of \(2+3 i ?\) What happens when you multiply this complex number by its complex conjugate?

In Exercises 59–94, solve each absolute value inequality. $$ |x+3| \leq 4 $$

Solve equation by completing the square. $$ 2 x^{2}-4 x-1=0 $$

In Exercises 61–78, solve each absolute value equation or indicate that the equation has no solution. $$|x|=6$$

A tennis club offers two payment options. Members can pay a monthly fee of \(\$ 30\) plus \(\$ 5\) per hour for court rental time. The second option has no monthly fee, but court time costs \(\$ 7.50\) per hour. a. Write a mathematical model representing total monthly costs for each option for \(x\) hours of court rental time. b. Use a graphing utility to graph the two models in a \([0,15,1]\) by \([0,120,20]\) viewing rectangle. c. Use your utility’s trace or intersection feature to determine where the two graphs intersect. Describe what the coordinates of this intersection point represent in practical terms. d. Verify part (c) using an algebraic approach by setting the two models equal to one another and determining how many hours one has to rent the court so that the two plans result in identical monthly costs.

Explaining the Concepts Explain how to plot a point in the rectangular coordinate system. Give an example with your explanation.

Solve each equation. Then determine whether the equation is an identity, a conditional equation, or an inconsistent equation. $$4 x+7=7(x+1)-3 x$$

Explain how to divide complex numbers. Provide an example with your explanation.

In Exercises 59–94, solve each absolute value inequality. $$ |2 x-6|<8 $$

Solve equation by completing the square. $$ 3 x^{2}-2 x-2=0 $$

In Exercises 61–78, solve each absolute value equation or indicate that the equation has no solution. $$|x-2|=7$$

Explaining the Concepts Explain why \((5,-2)\) and \((-2,5)\) do not represent the same point.

Solve each equation. Then determine whether the equation is an identity, a conditional equation, or an inconsistent equation. $$3(x+2)=7+3 x$$

In Exercises 59–94, solve each absolute value inequality. $$ |3 x+5|<17 $$

Solve equation by completing the square. $$ 3 x^{2}-5 x-10=0 $$

In Exercises 61–78, solve each absolute value equation or indicate that the equation has no solution. $$|x+1|=5$$

Explain how to graph an equation in the rectangular coordinate system.

Solve each equation. Then determine whether the equation is an identity, a conditional equation, or an inconsistent equation. $$4(x+5)=21+4 x$$

A stand-up comedian uses algebra in some jokes, including one about a telephone recording that announces "You have just reached an imaginary number. Please multiply by \(i\) and dial again." Explain the joke.

In Exercises 59–94, solve each absolute value inequality. $$ |2(x-1)+4| \leq 8 $$

Solve equation using the quadratic formula. $$ x^{2}+8 x+15=0 $$

In Exercises 61–78, solve each absolute value equation or indicate that the equation has no solution. $$|2 x-1|=5$$

Explaining the Concepts What does a \([-20,2,1]\) by \([-4,5,0.5]\) viewing rectangle mean?

Solve each equation. Then determine whether the equation is an identity, a conditional equation, or an inconsistent equation. $$ 10 x+3=8 x+3 $$

In Exercises 59–94, solve each absolute value inequality. $$ |3(x-1)+2| \leq 20 $$