Absolute Value Equations and Inequalities Word Problems

Absolute Value Equations and Inequalities Word Problems

By Callie Warren

Regular Inequalities

Does not need absolute value if there is no range

Yellow Cab Taxi charges a $1.75 flat rate in addition to $0.65 per mile.  Katie has no more than $10 to spend on a ride.​ "no more than" means "less than or equal to"

​

How many miles can Katie ride without exceeding her limit?​

​1.75 + 0.65x ≤ 10

0.65x ≤​ 8.25 x ≤ 12.7

​

​

1.75 + 0.65x ≤ 10

Regular Inequalities

Does not need absolute value if there is no range

​Ezra received $50 from his grandparents for his birthday. He makes $7.50 each week for odd jobs he does around the neighborhood.  Since his birthday, he has saved more than enough to  buy the $120 gift he wants to buy for his parents'  20th wedding anniversary. ​

50 + 7.50x >​ 120​

How many weeks should he work?

7.50x >​ 70 x > 9.3

Open Ended

Chris wants to order DVD's over the internet. Each DVD costs $15.99 and shipping the entire order costs $9.99. He can spend no more than $100.

Write the inequality.

How many DVDs can he buy?

**use <= or >= for those symbols if needed*

Chris wants to order DVD's over the internet. Each DVD costs $15.99 and shipping the entire order costs $9.99. He can spend no more than $100.

​

Write the inequality. 15.99x + 9.99 ≤ 100

How many DVDs can he buy?

15.99x ≤ 90.01

x ≤​ 5.62 which means he can only buy 5 DVDs

Multiple Choice

Melissa wants to spend no more than $300 on school clothes. She spends $75 on a coat and then wants to buy some sweaters that are on special for $10 each. Write an inequality to show how many sweaters Melissa can buy. DO NOT SOLVE

• 75x-10 ≥ 300
• 75+10 ≤ 300x
• 10x ≥ 300
• 75+10x ≤ 300

Multiple Choice

Lisa wants to change her cell phone service provider. She can spend no more than $80 each month. If the plan she is considering charges a flat fee of $57 and charges $10 per gigabyte of data, how many gigs of data could she include in her plan? 

• $2.30
• 2.3 gigs
• 1.2 gigs
• $1.22

Multiple Choice

Olivia is going to the mall where she will buy a coffee for $3.00 and soaps for $4.00 each. How many bars of soap can she buy if she will spend no more than $30.00?

• 6.75 bars
• 6 bars
• 7 bars
• 5 bars

Absolute Value Inequalities

Used when there is a range or margin ​

The normal human body temperature is 98.6 degrees. A temperature, x, is considered unhealthy if it differs more than 2 degrees from normal. Write the inequality showing the temperature for an unhealthy individual.

​Since the temperature can differ by 2 degrees, this requires absolute value bars.

The "normal" or "exact" goes inside the bars. The variation goes with the inequality symbol.

Absolute Value Inequalities

Used when there is a range or margin ​

The normal human body temperature is 98.6 degrees. A temperature, x, is considered unhealthy if it differs more than 2 degrees from normal. Write the inequality showing the temperature for an unhealthy individual.

Find the temperatures for an unhealthy individual.​

Absolute Value Inequalities

Used when there is a range or margin ​

A golf ball manufacture puts 390 golf balls in each box.  The box can be off by 6 golf balls.  Write and  solve an absolute value inequality that displays the  acceptable amount of golf balls that  can be in a box

Multiple Choice

The rules for an essay contest say that entries can have 500 words with an absolute deviation of at most 30 words. Which inequality represents represents the acceptable number of words?

• |x - 30| < 500

• |x - 30| ≤ 500

• |x - 500| < 30

• |x - 500| ≤ 30

Multiple Choice

A company makes boxes of crackers that should weigh 213g. A quality control inspector randomly selects boxes to weigh.  Any box that varies from the weight by more than 5g is sent back. What is an equation that shows the range of allowable weights for a box of crackers and what is the range?

• |w - 213| ≤ 5,         208 ≤ w ≤ 218
• |w - 213| ≤ 5,                   0 ≤ w ≤ 10
• |w - 5| ≤ 213,         208 ≤ w ≤ 218
• |w - 5| ≤ 213,                   0 ≤ w ≤ 10

Multiple Choice

The radius of the gears produced at a factory must be 6 inches in length with a tolerance of 0.1 inches. The gears with radius beyond the tolerated lengths will be thrown away. Which of the following inequalities can be used to assess which gears are eligible? (x is the length of the radius)

• |x – 6| ≤ 0.1
• |x – 6| ≥ 0.1
• |x – 0.1| ≤ 6
• |x – 0.1| ≥ 6

Multiple Choice

John is looking for a job after graduation. The salary which he is satisfied with must be $3000 with a tolerance of $500. Which of the following inequalities can be used to assess which if his salary is tolerable? (m is the measure of the salary)

• |m - 500| ≤ 3000
• |m - 500| ≥ 3000
• |m - 3000| ≥ 500
• |m - 3000| ≤ 500

Multiple Choice

John is looking for a job after graduation. The salary which he is satisfied with must be $3000 with a tolerance of $500. Choose the salary John would accept.

• $3,478
• $3,792
• $2,489
• $2,256

Absolute Value Equation

Used when there is a definite minimum and maximum

You have money in your wallet, but you don't know the exact amount. When a friend asks you, you say that you have 50 dollars give or take 15. Use an absolute value equation to find least and biggest amount of money in your pocket.

You either have $65 or $35 in your pocket, no in between.

Absolute Value Equation

Used when there is a definite minimum and maximum

A machine is used to fill each of several bags with 16 ounces of sugar. After the bags are filled, another machine weighs them. If the bag weighs .3 ounces more or less than the desired weight, the bag is rejected. Write this equation to find the heaviest and lightest bag the machine will approve.

The lightest bag can weigh 15.7 ounces and the heaviest bag can be 16.3 ounces.

Multiple Choice

Sam's house is 14 blocks from the school. Emily lives 6 blocks from Sam. Use an absolute value equation to determine Emily's distance for the school.

• |x + 6| = 10
  6, 14
• |x – 6| = 14
  20, 8
• |x – 14| = 6
  20, 8
• |x + 14| = 6
  6, 14

Multiple Choice

The average January temperature in a northern Canadian city is 1 degree Fahrenheit. The actual January temperature for that city may be about 5 degrees warmer or colder. Write an equation to find the minimum and maximum temperatures.

• |t - 1|=5

• |t - 5|=1

• |t + 1|=5

• |t + 5|=1

Multiple Choice

Sofia is buying new skis. She finds that the average price of skis is $750 but the actual price varies by $250. Write an equation to describe this situation.

• |x - 750|=250

• |x - 250|=750

• |x + 750|=250

• |x + 250|=750