Chapter 2

Solve each inequality and express the solution set using interval notation. \(7(x+1)-8(x-2)<0\)

Solve each of Problems \(47-62\) by setting up. How many milliliters of pure acid must be added to 150 milliliters of a \(30 \%\) solution of acid to obtain a \(40 \%\) solution?

Solve each equation and express the solutions in decimal form. Be sure to check your solutions. Use your calculator whenever it seems helpful. \(0.6(d-4.8)=7.38\)

Why must potential answers to word problems be checked back into the original statement of the problem?

Use an algebraic approach to solve each problem. Suppose that Maria has 150 coins consisting of pennies, nickels, and dimes. The number of nickels she has is 10 less than twice the number of pennies; the number of dimes she has is 20 less than three times the number of pennies. How many coins of each kind does she have?

Solve each equation and inequality by inspection. \(|x+9|>-6\)

Solve each problem by setting up and solving an appropriate inequality. Candace had scores of \(95,82,93\), and 84 on her first four exams of the semester. What score must she obtain on the fifth exam to have an average of 90 or better for the five exams?

Solve each inequality and express the solution set using interval notation. \(5(x-6)-6(x+2)<0\)

Solve each of Problems \(47-62\) by setting up. A 16 -quart radiator contains a \(50 \%\) solution of antifreeze. How much needs to be drained out and replaced with pure antifreeze to obtain a \(60 \%\) antifreeze solution?

Solve each equation and express the solutions in decimal form. Be sure to check your solutions. Use your calculator whenever it seems helpful. \(0.8(2 x-1.4)=19.52\)

Suppose your friend solved the problem, find two consecutive odd integers whose sum is 28 like this: $$ \begin{aligned} x+x+1 &=28 \\ 2 x &=27 \\ x &=\frac{27}{2}=13 \frac{1}{2} \end{aligned} $$ She claims that \(13 \frac{1}{2}\) will check in the equation. Where has she gone wrong and how would you help her?

Use an algebraic approach to solve each problem. Hector has a collection of nickels, dimes, and quarters totaling 122 coins. The number of dimes he has is 3 more than four times the number of nickels, and the number of quarters he has is 19 less than the number of dimes. How many coins of each kind does he have?

Solve each equation and inequality by inspection. \(|x+4|<0\)

Solve each problem by setting up and solving an appropriate inequality. Suppose that Derwin shot rounds of \(82,84,78\), and 79 on the first four days of a golf tournament. What must he shoot on the fifth day of the tournament to average 80 or less for the five days?

Solve each inequality and express the solution set using interval notation. \(5(x-6)-6(x+2)<0\)

Solve each of Problems \(47-62\) by setting up. Some people subtract 32 and then divide by 2 to estimate the change from a Fahrenheit reading to a Celsius reading. Why does this give an estimate and how good is the estimate?

Solve each equation and express the solutions in decimal form. Be sure to check your solutions. Use your calculator whenever it seems helpful. \(0.5(3 x+0.7)=20.6\)

Use an algebraic approach to solve each problem. The selling price of a ring is \(\$ 750\). This represents \(\$ 150\) less than three times the cost of the ring. Find the cost of the ring.

Solve each equation and inequality by inspection. \(|x+6|>0\)

Solve each problem by setting up and solving an appropriate inequality. The temperatures for a 24-hour period ranged between \(-4^{\circ} \mathrm{F}\) and \(23^{\circ} \mathrm{F}\), inclusive. What was the range in Celsius degrees? \(\left(\right.\) Use \(\mathrm{F}=\frac{9}{5} \mathrm{C}+32\).)

Solve each inequality and express the solution set using interval notation. \(3(x+2)+4<-2 x+14+x\)

Use an algebraic approach to solve each problem. In a class of 62 students, the number of females is one less than twice the number of males. How many females and how many males are there in the class?

Solve each equation and inequality by inspection. Explain how you would solve the inequality $$ |2 x+5|>-3 \text {. } $$

Solve each problem by setting up and solving an appropriate inequality. A person's intelligence quotient \((I)\) is found by dividing mental age \((M)\), as indicated by standard tests, by chronological age \((C)\) and then multiplying this ratio by 100 . The formula \(I=\frac{100 M}{C}\) can be used. If the \(I\) range of a group of 11-year-olds is given by \(80 \leq I \leq 140\), find the range of the mental age of this group.

Solve each inequality and express the solution set using interval notation. \(3(x-2)-5(2 x-1) \geq 0\)

Solve each equation and express the solutions in decimal form. Be sure to check your solutions. Use your calculator whenever it seems helpful. A retailer buys an item for \(\$ 90\), resells it for \(\$ 100\), and claims that she is making only a \(10 \%\) profit. Is this claim correct?

Use an algebraic approach to solve each problem. An apartment complex contains 230 apartments each having one, two, or three bedrooms. The number of two-bedroom apartments is 10 more than three times the number of three-bedroom apartments. The number of one-bedroom apartments is twice the number of twobedroom apartments. How many apartments of each kind are in the complex?

Why is 2 the only solution for \(|x-2| \leq 0\) ?

Solve each inequality and express the solution set using interval notation. \(4(2 x-1)-3(3 x+4) \geq 0\)

Use your calculator to help solve each formula for the indicated variable. Solve \(i=\) Prt for \(i\), given that \(P=\$ 875, r=12 \frac{1}{2} \%\), and \(t=4\) years.

Solve each equation and express the solutions in decimal form. Be sure to check your solutions. Use your calculator whenever it seems helpful. Is a \(10 \%\) discount followed by a \(20 \%\) discount equal to a \(30 \%\) discount? Defend your answer.

Use an algebraic approach to solve each problem. Barry sells bicycles on a salary-plus-commission basis. He receives a monthly salary of \(\$ 300\) and a commission of \(\$ 15\) for each bicycle that he sells. How many bicycles must he sell in a month to have a total monthly income of \(\$ 750\) ?

Explain how you would solve the equation \(|2 x-3|=0\).

Solve each inequality and express the solution set using interval notation. \(-5(3 x+4)<-2(7 x-1)\)

Solve \(i=\) Prt for \(i\), given that \(P=\$ 1125, r=13 \frac{1}{4} \%\), and \(t=4\) years.

Use an algebraic approach to solve each problem. Explain the difference between a numerical statement and an algebraic equation.

For Problems \(68-73\), solve each equation. \(|3 x+1|=|2 x+3|\)

Solve each problem by setting up and solving an appropriate inequality. Explain the difference between a conjunction and a disjunction. Give an example of each (outside the field of mathematics).

Solve each inequality and express the solution set using interval notation. \(-3(2 x+1)>-2(x+4)\)

Use an algebraic approach to solve each problem. Are the equations \(7=9 x-4\) and \(9 x-4=7\) equivalent equations? Defend your answer.

Solve each equation. \(|-2 x-3|=|x+1|\)

Solve each problem by setting up and solving an appropriate inequality. How do you know by inspection that the solution set of the inequality \(x+3>x+2\) is the entire set of real numbers?

Solve each inequality and express the solution set using interval notation. \(-3(x+2)>2(x-6)\)

Solve \(i=\) Prt for \(t\), given that \(i=\$ 243.75, P=\$ 1250\), and \(r=13 \%\).

Solve each equation. \(|2 x-1|=|x-3|\)

Solve each problem by setting up and solving an appropriate inequality. Find the solution set for each of the following compound statements, and in each case explain your reasoning. (a) \(x<3\) and \(5>2\) (b) \(x<3\) or \(5>2\) (c) \(x<3\) and \(6<4\) (d) \(x<3\) or \(6<4\)

Solve each inequality and express the solution set using interval notation. \(-2(x-4)<5(x-1)\)

Solve \(i=\operatorname{Prt}\) for \(r\), given that \(i=\$ 356.50, P=\$ 1550\), and \(t=2\) years. Express \(r\) as a percent.

Use an algebraic approach to solve each problem. Explain in your own words what it means to declare a variable when solving a word problem.

Solve each equation. \(|x-2|=|x+6|\)