Chapter 1

Absolute value expressions are equal when the expressions inside the absolute value bars are equal to or opposites of each other. $$ |2 x-7|=|x+3| $$

In Exercises \(57-76,\) solve each formula for the specified variable. Do you recognize the formula? If so, what does it describe? \(A=2 l w+2 l h+2 w h\) for \(h\)

Solve each inequality in Exercises 57-84 by first rewriting each one as an equivalent inequality without absolute value bars. Graph the solution set on a number line. Express the solution set using interval notation. $$\left|\frac{3 x-3}{9}\right| \geq 1$$

The equation \(P=-0.5 d+100\) describes the percentage, \(P,\) of lost hikers found in search and rescue missions when members of the search team walk parallel to one another separated by a distance of \(d\) yards. If a search and rescue team finds \(70 \%\) of lost hikers, find the parallel distance of separation between members of the search party.

Solve each equation in Exercises \(73-98\) by the method of your choice. \(5 x^{2}+2=11 x\)

Solve each equation by the method of your choice. $$ x+2 \sqrt{x}-3=0 $$

In Exercises \(57-76,\) solve each formula for the specified variable. Do you recognize the formula? If so, what does it describe? \(\frac{1}{p}+\frac{1}{q}=\frac{1}{f}\) for \(f\)

Solve each inequality in Exercises 57-84 by first rewriting each one as an equivalent inequality without absolute value bars. Graph the solution set on a number line. Express the solution set using interval notation. $$\left|3-\frac{2}{3} x\right|>5$$

What is a linear equation in one variable? Give an example of this type of equation.

Solve each equation in Exercises \(73-98\) by the method of your choice. \(5 x^{2}=6-13 x\)

Solve each equation by the method of your choice. $$ x^{3}+3 x^{2}-4 x-12=0 $$

Solve each inequality in Exercises 57-84 by first rewriting each one as an equivalent inequality without absolute value bars. Graph the solution set on a number line. Express the solution set using interval notation. $$\left|3-\frac{3}{4} x\right|>9$$

What does it mean to solve an equation?

Solve each equation in Exercises \(73-98\) by the method of your choice. \(3 x^{2}=60\)

Solve each equation by the method of your choice. $$ (x+4)^{3 / 2}=8 $$

What is a formula?

Solve each inequality by first rewriting each one as an equivalent inequality without absolute value bars. Graph the solution set on a number line. Express the solution set using interval notation. $$3|x-1|+2 \geq 8$$

What is the solution set of an equation?

Solve each equation in Exercises \(73-98\) by the method of your choice. \(2 x^{2}=250\)

Solve each equation by the method of your choice. $$ \left(x^{2}-1\right)^{2}-2\left(x^{2}-1\right)=3 $$

We discussed formulas in this section after we considered procedures for solving linear equations. Doesn't working -with a formula simply mean substituting given numbers Jinto the formula and using the order of operations? Is it necessary to know how to solve equations to work with formulas? Explain.

Solve each inequality by first rewriting each one as an equivalent inequality without absolute value bars. Graph the solution set on a number line. Express the solution set using interval notation. $$-2|4-x| \geq-4$$

What are equivalent equations? Give an example.

Solve each equation in Exercises \(73-98\) by the method of your choice. \(x^{2}-2 x=1\)

Solve each equation by the method of your choice. $$ \sqrt{4 x+15}-2 x=0 $$

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

Solve each inequality by first rewriting each one as an equivalent inequality without absolute value bars. Graph the solution set on a number line. Express the solution set using interval notation. $$3<|2 x-1|$$

What is the difference between solving an equation such as \(2(x-4)+5 x=34\) and simplifying an algebraic expression such as \(2(x-4)+5 x ?\) If there is a difference, which topic should be taught first? Why?

Solve each equation in Exercises \(73-98\) by the method of your choice. \(2 x^{2}+3 x=1\)

Solve each equation by the method of your choice. $$ x^{2 / 5}-1=0 $$

Did you have some 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?

Solve each inequality by first rewriting each one as an equivalent inequality without absolute value bars. Graph the solution set on a number line. Express the solution set using interval notation. $$5 \geq|4-x|$$

Suppose that you solve \(\frac{x}{5}-\frac{x}{2}=1\) by multiplying both sides by \(20,\) rather than the least common denominator of 5 and 2 (namely, 10 ). Describe what happens. If you get the correct solution, why do you think we clear the equation of fractions by multiplying by the least common denominator?

Solve each equation in Exercises 73-98 by the method of your choice. \((2 x+3)(x+4)=1\)

Solve each equation by the method of your choice. $$ \left|x^{2}+2 x-36\right|=12 $$

Solve each inequality by first rewriting each one as an equivalent inequality without absolute value bars. Graph the solution set on a number line. Express the solution set using interval notation. $$12<\left|-2 x+\frac{6}{7}\right|+\frac{3}{7}$$

A. Suppose you are an algebra teacher grading the following solution on an examination: $$\begin{aligned} -3(x-6) &=2-x \\ -3 x-18 &=2-x \\ -2 x-18 &=2 \\ -2 x &=-16 \\ x &=8 \end{aligned}$$ You should note that 8 checks, and the solution set is \(\\{8\\} .\) The student who worked the problem therefore wants full credit. Can you find any errors in the solution? If full credit is 10 points, how many points should you give the student? Justify your position.

Solve each equation in Exercises 73-98 by the method of your choice. \((2 x-5)(x+1)=2\)

Solve each equation by the method of your choice. $$ \sqrt{3 x+1}-\sqrt{x-1}=2 $$

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 \(\mathbf{a}[0,15,1]\) 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.

Solve each inequality by first rewriting each one as an equivalent inequality without absolute value bars. Graph the solution set on a number line. Express the solution set using interval notation. $$1<\left|x-\frac{11}{3}\right|+\frac{7}{3}$$

Explain how to determine the restrictions on the variable for the equation $$ \frac{3}{x+5}+\frac{4}{x-2}=\frac{7}{(x+5)(x-2)} $$

Solve each equation in Exercises 73-98 by the method of your choice. \((3 x-4)^{2}=16\)

Solve each equation by the method of your choice. $$ x^{3}-2 x^{2}=x-2 $$

Solve each inequality by first rewriting each one as an equivalent inequality without absolute value bars. Graph the solution set on a number line. Express the solution set using interval notation. $$4+\left|3-\frac{x}{3}\right| \geq 9$$

What is an identity? Give an example.

Solve each equation in Exercises 73-98 by the method of your choice. \((2 x+7)^{2}=25\)

Solve each equation by the method of your choice. $$ \left|x^{2}+6 x+1\right|=8 $$

Solve each inequality by first rewriting each one as an equivalent inequality without absolute value bars. Graph the solution set on a number line. Express the solution set using interval notation. $$\left|2-\frac{x}{2}\right|-1 \leq 1$$

What is a conditional equation? Give an example.