Chapter 1

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

In Exercises \(75-82\), compute the discriminant. Then determine the number and type of solutions for the given equation. $$x^{2}-4 x-5=0$$

Perform the indicated operations and write the result in standard form. $$ \frac{4}{(2+i)(3-i)} $$

List the quadrant or quadrants satisfying each condition. $$x y>0$$

Solve each absolute value equation or indicate that the equation has no solution. $$|2 x-1|+3-3$$

Solve absolute value inequality. \(\left|\frac{3 x-3}{9}\right| \geq 1\)

combine the types of equations we have discussed in this section. Solve each equation. Then state whether the equation is an identity, a conditional equation, or an inconsistent equation. $$ \frac{3}{x}-\frac{1}{6}=\frac{1}{3} $$

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

In Exercises \(75-82\), compute the discriminant. Then determine the number and type of solutions for the given equation. $$4 x^{2}-2 x+3=0$$

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

List the quadrant or quadrants satisfying each condition. $$\frac{y}{x}<0$$

Solve each absolute value equation or indicate that the equation has no solution. $$|3 x-2|+4-4$$

Solve absolute value inequality. \(\left|3-\frac{2}{3} x\right|>5\)

Explain what it means to solve a formula for a variable.

In Exercises \(75-82\), compute the discriminant. Then determine the number and type of solutions for the given equation. $$2 x^{2}-11 x+3=0$$

Perform the indicated operations and write the result in standard form. $$ \frac{8}{1+\frac{2}{i}} $$

List the quadrant or quadrants satisfying each condition. $$x^{3}>0 \text { and } y^{3}<0$$

Solve absolute value inequality. \(\left|3-\frac{3}{4} x\right|>9\)

combine the types of equations we have discussed in this section. Solve each equation. Then state whether the equation is an identity, a conditional equation, or an inconsistent equation. $$ \frac{1}{x-1}=\frac{1}{(2 x+3)(x-1)}+\frac{4}{2 x+3} $$

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?

In Exercises \(75-82\), compute the discriminant. Then determine the number and type of solutions for the given equation. $$2 x^{2}+11 x-6=0$$

Exercises \(78-80\) will help you prepare for the material covered in the next section. Factor: $$ 2 x^{2}+7 x-4 $$

List the quadrant or quadrants satisfying each condition. $$x^{3}<0 \text { and } y^{3}>0$$

Solve absolute value inequality. \(3|x-1|+2 \geq 8\)

combine the types of equations we have discussed in this section. Solve each equation. Then state whether the equation is an identity, a conditional equation, or an inconsistent equation. $$ \frac{4 x}{x+3}-\frac{12}{x-3}=\frac{4 x^{2}+36}{x^{2}-9} $$

In Exercises \(75-82\), compute the discriminant. Then determine the number and type of solutions for the given equation. $$x^{2}-2 x+1=0$$

Exercises \(78-80\) will help you prepare for the material covered in the next section. Factor: $$ x^{2}-6 x+9 $$

Solve absolute value inequality. \(5|2 x+1|-3 \geq 9\)

combine the types of equations we have discussed in this section. Solve each equation. Then state whether the equation is an identity, a conditional equation, or an inconsistent equation. $$ \frac{4}{x^{2}+3 x-10}-\frac{1}{x^{2}+x-6}=\frac{3}{x^{2}-x-12} $$

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.

In Exercises \(75-82\), compute the discriminant. Then determine the number and type of solutions for the given equation. $$3 x^{2}-2 x=1$$

Solve absolute value inequality. \(-2|x-4| \geq-4\)

In Exercises \(75-82\), compute the discriminant. Then determine the number and type of solutions for the given equation. $$x^{2}-3 x-7=0$$

Solve absolute value inequality. \(-3|x+7| \geq-27\)

In Exercises \(75-82\), compute the discriminant. Then determine the number and type of solutions for the given equation. $$3 x^{2}+4 x-2=0$$

Solve absolute value inequality. \(-4|1-x|<-16\)

Solve each equation in Exercises \(83-108\) by the method of your choice. $$2 x^{2}-x=1$$

Solve absolute value inequality. \(-2|5-x|<-6\)

Solve each equation in Exercises \(83-108\) by the method of your choice. $$3 x^{2}-4 x=4$$

Solve absolute value inequality. \(3 \leq|2 x-1|\)

Evaluate \(x^{2}-x\) for the value of \(x\) satisfying \(4(x-2)+2=4 x-2(2-x)\)

Solve each equation in Exercises \(83-108\) by the method of your choice. $$5 x^{2}+2=11 x$$

Solve absolute value inequality. \(9 \leq|4 x+7|\)

Evaluate \(x^{2}-x\) for the value of \(x\) satisfying \(2(x-6)=3 x+2(2 x-1)\)

Solve each equation in Exercises \(83-108\) by the method of your choice. $$5 x^{2}=6-13 x$$

Solve absolute value inequality. \(5>|4-x|\)

Evaluate \(x^{2}-(x y-y)\) for \(x\) satisfying \(\frac{3(x+3)}{5}=2 x+6\) and \(y\) satisfying \(-2 y-10=5 y+18\)

In a film, the actor Charles Coburn plays an elderly "uncle" character criticized for marrying a woman when he is 3 times her age. He wittily replies, "Ah, but in 20 years time I shall only be twice her age." How old is the "uncle" and the woman?

This will help you prepare for the material covered in the next section. If 6 is substituted for \(x\) in the equation $$2(x-3)-17=13-3(x+2)$$ is the resulting statement true or false?

Solve absolute value inequality. \(2>|1-x|\)