Chapter 1

The equations in Exercises \(59-70\) combine the types of equations we have discussed in this section. Solve each equation or state that it is true for all real numbers or no real numbers. $$ \frac{2}{x}+\frac{1}{2}=\frac{3}{4} $$

Which one of the following is true? a. The solution set of \(x^{2}>25\) is \((5, \infty)\) b. The inequality \(\frac{x-2}{x+3}<2\) can be solved by multiplying both sides by \(x+3\), resulting in the equivalent inequality \(x-2<2(x+3)\) c. \((x+3)(x-1) \geq 0\) and \(\frac{x+3}{x-1} \geq 0\) have the same solution set. d. None of these statements is true.

Solve each absolute value equation or indicate the equation has no solution. $$ 3|2 x-1|=21 $$

Compute the discriminant of each equation in Exercises 65-72 What does the discriminant indicate about the number and type of solutions? \(4 x^{2}-2 x+3=0\)

In Exercises \(57-76,\) solve each formula for the specified variable. Do you recognize the formula? If so, what does it describe? \(P=C+M C\) for \(M\)

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. $$\left|\frac{3(x-1)}{4}\right|<6$$

The equations in Exercises \(59-70\) combine the types of equations we have discussed in this section. Solve each equation or state that it is true for all real numbers or no real numbers. $$ \frac{3}{x}-\frac{1}{6}=\frac{1}{3} $$

Write a quadratic inequality whose solution set is \([-3,5]\)

Compute the discriminant of each equation in Exercises 65-72 What does the discriminant indicate about the number and type of solutions? \(2 x^{2}-11 x+3=0\)

Solve each absolute value equation or indicate the equation has no solution. $$ 7|5 x|+2=16 $$

In Exercises \(57-76,\) solve each formula for the specified variable. Do you recognize the formula? If so, what does it describe? \(A=\frac{1}{2} h(a+b)\) for \(a\)

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. $$|x|>3$$

The equations in Exercises \(59-70\) combine the types of equations we have discussed in this section. Solve each equation or state that it is true for all real numbers or no real numbers. $$ \frac{4}{x-2}+\frac{3}{x+5}=\frac{7}{(x+5)(x-2)} $$

Write a rational inequality whose solution set is \((-\infty,-4)\) or \([3, \infty)\)

Compute the discriminant of each equation in Exercises 65-72 What does the discriminant indicate about the number and type of solutions? \(2 x^{2}+11 x-6=0\)

Solve each absolute value equation or indicate the equation has no solution. $$ 7|3 x|+2=16 $$

In Exercises \(57-76,\) solve each formula for the specified variable. Do you recognize the formula? If so, what does it describe? \(A=\frac{1}{2} h(a+b)\) for b

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. $$|x|>5$$

The equations in Exercises \(59-70\) combine the types of equations we have discussed in this section. Solve each equation or state that it is true for all real numbers or no real numbers. $$ \frac{1}{x-1}=\frac{1}{(2 x+3)(x-1)}+\frac{4}{2 x+3} $$

In Exercises \(69-72,\) use inspection to describe each inequality's solution set. Do not solve any of the inequalities. $$ (x-2)^{2}>0 $$

Compute the discriminant of each equation in Exercises 65-72 What does the discriminant indicate about the number and type of solutions? \(x^{2}-2 x+1=0\)

Solve each absolute value equation or indicate the equation has no solution. $$ |x+1|+5=3 $$

In Exercises \(57-76,\) solve each formula for the specified variable. Do you recognize the formula? If so, what does it describe? \(S=P+P r t\) for \(r\)

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. $$|x-1| \geq 2$$

The equations in Exercises \(59-70\) combine the types of equations we have discussed in this section. Solve each equation or state that it is true for all real numbers or no real numbers. $$ \frac{4 x}{x+3}-\frac{12}{x-3}=\frac{4 x^{2}+36}{x^{2}-9} $$

In Exercises \(69-72,\) use inspection to describe each inequality's solution set. Do not solve any of the inequalities. $$ (x-2)^{2} \leq 0 $$

Compute the discriminant of each equation in Exercises 65-72 What does the discriminant indicate about the number and type of solutions? \(3 x^{2}=2 x-1\)

Solve each absolute value equation or indicate the equation has no solution. $$ |x+1|+6=2 $$

In Exercises \(57-76,\) solve each formula for the specified variable. Do you recognize the formula? If so, what does it describe? \(S=P+P r t\) for \(t\)

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. $$|x+3| \geq 4$$

The equations in Exercises \(59-70\) combine the types of equations we have discussed in this section. Solve each equation or state that it is true for all real numbers or no real numbers. $$ \frac{4}{x^{2}+3 x-10}-\frac{1}{x^{2}+x-6}=\frac{3}{x^{2}-x-12} $$

Compute the discriminant of each equation in Exercises 65-72 What does the discriminant indicate about the number and type of solutions? \(x^{2}-3 x-7=0\)

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

In Exercises \(57-76,\) solve each formula for the specified variable. Do you recognize the formula? If so, what does it describe? \(B=\frac{F}{S-V}\) for \(S\)

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. $$|3 x-8|>7$$

The equation \(d=5000 c-525,000\) describes the relationship between the annual number of deaths, \(d\), in the United States from heart disease and the average cholesterol level, \(c,\) of blood. (Cholesterol level, \(c,\) is expressed in milligrams per deciliter of blood.) a. In \(2000,725,000\) Americans died from heart disease. Substitute \(725,000\) for \(d\) in the given equation and then solve for \(c\) to determine the average cholesterol level in 2000 . b. Suppose that the average cholesterol level for people in the United States could be reduced to 180 . Substitute 180 for \(c\) in the given equation and then compute the value for \(d\) to determine the number of annual deaths from heart disease with this reduced cholesterol level. Compared to the number of deaths in \(2000,\) how many lives would be saved by this cholesterol reduction?

In Exercises \(69-72,\) use inspection to describe each inequality's solution set. Do not solve any of the inequalities. $$ \frac{1}{(x-2)^{2}}>0 $$

Compute the discriminant of each equation in Exercises 65-72 What does the discriminant indicate about the number and type of solutions? \(3 x^{2}+4 x-2=0\)

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

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

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. $$|5 x-2|>13$$

There is a relationship between the vocabulary of a child and the child's age. The equation \(60 A-V=900\) describes this relationship, where \(A\) is the age of the child, in months, and \(V\) is the number of words that the child uses. Suppose that a child uses 1500 words. Determine the child's age, in months.

In Exercises \(73-74\), use the method for solving quadratic inequalities to solve each higher-order polynomial inequality. $$ x^{3}+x^{2}-4 x-4>0 $$

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

Solve each absolute value equation or indicate the equation has no solution. $$ |3 x-1|=|x+5| $$

In Exercises \(57-76,\) solve each formula for the specified variable. Do you recognize the formula? If so, what does it describe? \(I R+I r=E\) for \(I\)

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. $$\left|\frac{2 x+2}{4}\right| \geq 2$$

The equation $$p=15+\frac{15 d}{33}$$ describes the pressure of sea water, \(p,\) in pounds per square foot, at a depth of \(d\) feet below the surface. The record depth for breath-held diving, by Francisco Ferreras (Cuba) off Grand-Bahama Island, on November \(14,1993\), involved pressure of 201 pounds per square foot. To what depth did Ferreras descend on this ill-advised venture? (He was underwater for 2 minutes and 9 seconds!)

In Exercises \(73-74\), use the method for solving quadratic inequalities to solve each higher-order polynomial inequality. $$ x^{3}+2 x^{2}-x-2 \geq 0 $$

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