To review these concepts, go to Inequality Word Problems Study Guide.

**Inequality Word Problems Practice Questions**

**Practice 1**

**Problems**

Using the list of phrases and key words from the chart, translate each of the following into mathematical symbols. Let *x* = a number in each exercise.

- Twenty is more than a number.
- The difference of a number and three is not more than two.
- The minimum value of a number is 65.
- Thirty times a number is at most 90.

**Solutions**

- 20 >
*x* *x*– 3 ≤ 2*x*≥ 65- 30
*x*≤ 90

**Practice 2**

**Problems**

Solve each inequality for *x*.

- x – 5 > 23
- < –9
- –3
*x*+ 4 ≤ 16 - 5(
*x*– 2) ≥ 7*x*

**Solutions**

- Add 5 to each side of the inequality:
*x*> 28. - Multiply each side of the inequality by 6:
*x*< –54. - Subtract 4 from each side of the inequality to get –3
*x*= 12. Divide each side by –3 and switch the inequality symbol:*x*= –4. - Use the distributive property on the left side of the inequality to get 5
*x*–10 = 7*x*. Subtract 5*x*from each side to get –10 = 2*x*. Divide each side of the inequality by 2: –5 =*x*.

**Practice 3**

**Problems**

- Four more than a number is at least 41. What is the least value of the number?
- Twice a number decreased by 7 is less than or equal to 21. What is the greatest value of the number?
- A number divided by –3 is at most –10. What is the minimum value of the number?
- Tyler and Melissa each have gumdrops. The amount that Tyler has is equal to five more than the amount that Melissa has. What is the minimum number of gumdrops that Tyler has if there are at least 31 gumdrops between them

**Solutions**

*Read and understand the question*. This question is looking for a number when clues about this number are given.*Read and understand the question*. This question is looking for a number when clues about this number are given.*Read and understand the question*. This question is looking for a number when clues about this number are given.*Read and understand the question*. This question is looking for the minimum number of gumdrops Tyler has. Tyler has five more than Melissa and there are 31 or more gumdrops between them.

*Make a plan*. Translate the statement into inequality form. Then, solve for *x* using the inequality solving steps.

*Carry out the plan*. Let *x* = a number. The key phrase *more than* means addition. The first part of the statement translates to *x* + 4. The key phrase *is at least* means is greater than or is equal to. The entire inequality is *x* + 4 = 41. Subtract 4 from each side of the inequality to get *x* = 37, so 37 is the least value of the number.

*Check your answer*. Check your solution by substituting the answer into the inequality.

*x*+ 4 = 41

becomes

- 37 + 4 = 41

- 41 = 41

This statement is true, so the solution is checking.

*Make a plan*. Translate the statement into inequality form. Then, solve for *x* using the inequality solving steps.

*Carry out the plan*. Let *x* = a number. The key phrase *decreased* by means subtraction, and twice a number is written as 2*x*. The first part of the statement translates to 2*x* –7. This amount is less than or equal to 21, so the entire inequality is 2*x* –7 = 21. Add 7 to each side of the inequality to get 2*x* = 28. Next, divide each side by 2 to get the variable alone.

*x*= 14

The greatest value of the number is 14.

*Check your answer*. Check your solution by substituting the answer into the inequality.

- 2

*x*– 7 = 21

becomes

- 2(14) – 7 = 21

- 28 – 7 = 21

- 21 = 21

This statement is true, so the solution is checking.

*Make a plan*. Translate the statement into inequality form. Then, solve for *x* using the inequality solving steps.

*Carry out the plan*. Let *x* = a number. The key phrase *is at most* means less than or equal to, so the statement translates to the inequality ≤ –10. Multiply each side by –3 and switch the direction of the inequality symbol.

- –3 × ≥ –10 × –3

*x*= 30

The minimum value is 30.

*Check your answer*. Check your solution by substituting the answer into the inequality.

- ≤ –10

becomes

- ≤ –10

- –10 = –10

This statement is true, so the solution is checking.

*Make a plan*. Translate the statement into inequality form. Then, solve for *x* using the inequality solving steps.

*Carry out the plan*. Let *x* = the number of gumdrops Melissa has, and let *x* + 5 = the number of gumdrops Tyler has. The key phrase *is at least* means is greater than or equal to, so add the amounts each person has to get the inequality *x* + *x* + 5 = 31. Combine like terms to simplify the inequality to 2*x* + 5 = 31. Subtract 5 from each side of the inequality to get 2*x* = 26. Next, divide each side by 2 to get the variable alone.

*x*= 13

Melissa has at least 13 gumdrops, and Tyler has at least 13 + 5 = 18 gumdrops.

*Check your answer*. Check your solution by adding the amounts each person has, to be sure this amount is at least 31: 13 + 18 = 31, so the solution is checking.

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