2.7 Solve Absolute Value Inequalities

Before you get started, take this readiness quiz.

Evaluate: $− \left|7\right| .$
If you missed this problem, review the reference.

Fill in $< , > ,$ or $=$ for each of the following pairs of numbers.
ⓐ $\left|−8\right| ___ - \left|−8\right|$ ⓑ $12 ___ - \left|−12\right|$ ⓒ $\left|−6\right| ___ - 6$ ⓓ $− \left(−15\right) ___ - \left|−15\right|$
If you missed this problem, review the reference.

Simplify: $14 - 2 \left|8 - 3 \left(4 - 1\right)\right| .$
If you missed this problem, review the reference.

Solve: ⓐ $\left|x\right| = 8$ ⓑ $\left|y\right| = −6$ ⓒ $\left|z\right| = 0$

Solve: ⓐ $\left|x\right| = 2$ ⓑ $\left|y\right| = −4$ ⓒ $\left|z\right| = 0$

Solve: ⓐ $\left|x\right| = 11$ ⓑ $\left|y\right| = −5$ ⓒ $\left|z\right| = 0$

Solve: $\left|3 x - 5\right| - 1 = 6 .$

Solve: $\left|4 x - 3\right| - 5 = 2 .$

Solve: $3 \left|x - 4\right| - 4 = 8 .$

Solve: $2 \left|x - 5\right| + 3 = 9 .$

Solve: $\left|\frac{2}{3} x - 4\right| + 11 = 3 .$

Solve: $\left|\frac{3}{4} x - 5\right| + 9 = 4 .$

Solve: $\left|\frac{5}{6} x + 3\right| + 8 = 6 .$

Solve: $\left|5 x - 1\right| = \left|2 x + 3\right| .$

Solve: $\left|7 x - 3\right| = \left|3 x + 7\right| .$

Solve: $\left|6 x - 5\right| = \left|3 x + 4\right| .$

The ideal diameter of a rod needed for a machine is 60 mm. The actual diameter can vary from the ideal diameter by $0.075$ mm. What range of diameters will be acceptable to the customer without causing the rod to be rejected?

The ideal diameter of a rod needed for a machine is 80 mm. The actual diameter can vary from the ideal diameter by 0.009 mm. What range of diameters will be acceptable to the customer without causing the rod to be rejected?

The ideal diameter of a rod needed for a machine is 75 mm. The actual diameter can vary from the ideal diameter by 0.05 mm. What range of diameters will be acceptable to the customer without causing the rod to be rejected?

Solve.

ⓐ $\left|x\right| = 4$ ⓑ $\left|y\right| = −5$ ⓒ $\left|z\right| = 0$

Solve.

ⓐ $\left|x\right| = 3$ ⓑ $\left|y\right| = −1$ ⓒ $\left|z\right| = 0$

Solve.

$\left|4 x - 1\right| - 3 = 0$

Solve.

$\left|4 x + 7\right| + 2 = 5$

Solve.

$3 \left|x - 4\right| + 2 = 11$

Solve.

$3 \left|x + 2\right| - 5 = 4$

Solve.

$−3 \left|x - 4\right| + 4 = −5$

Solve.

$\left|\frac{3}{5} x - 2\right| + 5 = 2$

Solve.

$\left|\frac{1}{4} x + 3\right| + 3 = 1$

Solve.

$\left|4 x + 3\right| = \left|2 x + 1\right|$

Solve.

$\left|6 - x\right| = \left|3 - 2 x\right|$

Solve. For each inequality, also graph the solution and write the solution in interval notation.

$\left|6 x - 5\right| = \left|2 x + 3\right|$

Solve. For each inequality, also graph the solution and write the solution in interval notation.

$\left|2 x - 5\right| + 2 = 3$

Solve. For each inequality, also graph the solution and write the solution in interval notation.

$\left|3 x + 1\right| - 3 = 7$

Solve. For each inequality, also graph the solution and write the solution in interval notation.

$5 \left|2 x - 1\right| - 3 = 7$

Solve.

An organic juice bottler ideally produces 215,000 bottle per day. But this total can vary by as much as 7,500 bottles. What is the maximum and minimum expected production at the bottling company?

Solve.

At Lilly’s Bakery, the ideal weight of a loaf of bread is 24 ounces. By law, the actual weight can vary from the ideal by 1.5 ounces. What range of weight will be acceptable to the inspector without causing the bakery being fined?

In your own words, explain how to solve the absolute value inequality, $\left|3 x - 2\right| \geq 4 .$

Determine whether each number is a solution to the equation.

$−12 n + 5 = 8 n , n = - \frac{5}{4}$

Solve each linear equation.

$− \left(s + 4\right) = 18$

Solve each linear equation.

$\frac{1}{3} \left(6 m + 21\right) = m - 7$

Solve each linear equation.

$0.25 \left(q - 8\right) = 0.1 \left(q + 7\right)$

Solve each linear equation.

$5 + 7 \left(2 - 5 x\right) = 2 \left(9 x + 1\right) - \left(13 x - 57\right)$

Solve each linear equation.

$2 \left[−16 + 5 \left(8 k - 6\right)\right] = 8 \left(3 - 4 k\right) - 32$

Classify each equation as a conditional equation, an identity, or a contradiction and then state the solution.

$9 u + 32 = 15 \left(u - 4\right) - 3 \left(2 u + 21\right)$

Solve each equation.

$\frac{2}{5} n - \frac{1}{10} = \frac{7}{10}$

Solve each equation.

$\frac{1}{2} \left(k + 3\right) = \frac{1}{3} \left(k + 16\right)$

Solve each equation.

$0.8 x - 0.3 = 0.7 x + 0.2$

Solve using the problem solving strategy for word problems.

Three-fourths of the people at a concert are children. If there are 87 children, what is the total number of people at the concert?

Solve each number word problem.

The sum of a number and three is forty-one. Find the number.

Solve each number word problem.

One number is two more than four times another. Their sum is negative thirteen. Find the numbers.

Solve each number word problem.

Find three consecutive even integers whose sum is 234.

Solve each number word problem.

Koji has $5,502 in his savings account. This is $30 less than six times the amount in his checking account. How much money does Koji have in his checking account?

Translate and solve.

12.5% of what number is 20?

Solve.

The bill for Dino’s lunch was $19.45. He wanted to leave 20% of the total bill as a tip. How much should the tip be?

Solve.

Jaden earns $2,680 per month. He pays $938 a month for rent. What percent of his monthly pay goes to rent?

Solve.

Rowena’s monthly gasoline bill dropped from $83.75 last month to $56.95 this month. Find the percent change.

Solve.

Lacey bought a pair of boots on sale for $95. The original price of the boots was $200. Find ⓐ the amount of discount and ⓑ the discount rate. (Round to the nearest tenth of a percent, if needed.)

Solve.

Winston deposited $3,294 in a bank account with interest rate 2.6% How much interest was earned in five years?

Solve.

Jaime’s refrigerator loan statement said he would pay $1,026 in interest for a four-year loan at 13.5%. How much did Jaime borrow to buy the refrigerator?

Solve the formula for the specified variable.

Solve the formula
$A = \frac{1}{2} d_{1} d_{2}$ for $d_{2} .$

Solve the formula for the specified variable.

Solve the formula
$4 x - 3 y = 12$ for y.

Solve using a geometry formula.

The measure of the smallest angle in a right triangle is $45 °$ less than the measure of the next larger angle. Find the measures of all three angles.

Solve using a geometry formula.

The length of a rectangle is 12 cm more than the width. The perimeter is 74 cm. Find the length and the width.

Solve using a geometry formula.

The perimeter of a triangle is 35 feet. One side of the triangle is five feet longer than the second side. The third side is three feet longer than the second side. Find the length of each side.

Solve.

Lenny has $3.69 in pennies, dimes, and quarters. The number of pennies is three more than the number of dimes. The number of quarters is twice the number of dimes. How many of each coin does he have?

Solve each ticket or stamp word problem.

125 tickets were sold for the jazz band concert for a total of $1,022. Student tickets cost $6 each and general admission tickets cost $10 each. How many of each kind of ticket were sold?

Solve.

Marquese is making 10 pounds of trail mix from raisins and nuts. Raisins cost $3.45 per pound and nuts cost $7.95 per pound. How many pounds of raisins and how many pounds of nuts should Marquese use for the trail mix to cost him $6.96 per pound?

Solve.

Enrique borrowed $23,500 to buy a car. He pays his uncle 2% interest on the $4,500 he borrowed from him, and he pays the bank 11.5% interest on the rest. What average interest rate does he pay on the total $23,500? (Round your answer to the nearest tenth of a percent.)

Solve.

Louellen and Tracy met at a restaurant on the road between Chicago and Nashville. Louellen had left Chicago and drove 3.2 hours towards Nashville. Tracy had left Nashville and drove 4 hours towards Chicago, at a speed one mile per hour faster than Louellen’s speed. The distance between Chicago and Nashville is 472 miles. Find Louellen’s speed and Tracy’s speed.

Solve.

Kyle rowed his boat upstream for 50 minutes. It took him 30 minutes to row back downstream. His speed going upstream is two miles per hour slower than his speed going downstream. Find Kyle’s upstream and downstream speeds.

Solve.

Anthony drove from New York City to Baltimore, which is a distance of 192 miles. He left at 3:45 and had heavy traffic until 5:30. Traffic was light for the rest of the drive, and he arrived at 7:30. His speed in light traffic was four miles per hour more than twice his speed in heavy traffic. Find Anthony’s driving speed in heavy traffic and light traffic.

Solve.

Julianne has a weekly food budget of $231 for her family. If she plans to budget the same amount for each of the seven days of the week, what is the maximum amount she can spend on food each day?

Solve.

Briana has been offered a sales job in another city. The offer was for $42,500 plus 8% of her total sales. In order to make it worth the move, Briana needs to have an annual salary of at least $66,500. What would her total sales need to be for her to move?

Solve.

Costa is an accountant. During tax season, he charges $125 to do a simple tax return. His expenses for buying software, renting an office, and advertising are $6,000. How many tax returns must he do if he wants to make a profit of at least $8,000?

Solve.

Elouise is creating a rectangular garden in her back yard. The length of the garden is 12 feet. The perimeter of the garden must be at least 36 feet and no more than 48 feet. Use a compound inequality to find the range of values for the width of the garden.

Solve.

$\left|y\right| = −14$

Solve.

$\left|3 x - 4\right| + 5 = 7$

Solve.

$−2 \left|x - 3\right| + 8 = −4$

Solve.

$\left|6 x - 5\right| = \left|2 x + 3\right|$

Solve.

A craft beer brewer needs 215,000 bottle per day. But this total can vary by as much as 5,000 bottles. What is the maximum and minimum expected usage at the bottling company?

Solve each equation.

$−5 \left(2 x + 1\right) = 45$

Solve each equation.

$8 \left(3 a + 5\right) - 7 \left(4 a - 3\right) = 20 - 3 a$

Solve each equation.

$14 n - 3 \left(4 n + 5\right) = −9 + 2 \left(n - 8\right)$

Solve each equation.

$\frac{3}{4} x - \frac{2}{3} = \frac{1}{2} x + \frac{5}{6}$

Solve each equation.

$\left|2 x - 1\right| = \left|4 x + 3\right|$

Translate to an equation or inequality and solve.

The sum of two consecutive odd integers is $−112 .$ Find the numbers.

Translate to an equation or inequality and solve.

Bonita has $2.95 in dimes and quarters in her pocket. If she has five more dimes than quarters, how many of each coin does she have?

Translate to an equation or inequality and solve.

The measure of one angle of a triangle is twice the measure of the smallest angle. The measure of the third angle is three times the measure of the smallest angle. Find the measures of all three angles.

Translate to an equation or inequality and solve.

Two planes leave Dallas at the same time. One heads east at a speed of 428 miles per hour. The other plane heads west at a speed of 382 miles per hour. How many hours will it take them to be 2,025 miles apart?

Translate to an equation or inequality and solve.

Sara has a budget of $1,000 for costumes for the 18 members of her musical theater group. If all the costumes are the same price, what is the maximum she can spend for each costume?