How do I solve word problems about inequality in math?

In an equation, both sides have the same value, so they are equal and stay balanced. An inequality is different.

An inequality compares two values rather than saying they are equal to show whether one value is greater than, less than, greater than or equal to, or less than or equal to the other.

While an equation uses an equals sign (=), an inequality uses symbols like > (greater than), < (less than), ≥ (greater than or equal to), or ≤ (less than or equal to). Unlike most equations, which usually have one solution, inequalities often have many possible solutions. Any number that makes the inequality true is part of the solution set.

When solving inequality word problems, you'll need to recognize clue words and phrases — such as at least, more than, or no more than — and translate them into the correct inequality symbols. Learning these key words will help you write and solve inequalities with confidence.

Watch for the phrases below when translating a word problem into a solvable equation: To hear an audio description of the chart above, click play on the audio player below:

Here are some sample translations:

• > means "more than"

  • The temperature is more than 75°F. → t > 75

• < means "Less than"

  • The ride lasts less than 3 minutes. → r < 3

• ≥ means "at least"

  • You must be at least 13 years old. → a ≥ 13

• ≤ means "at most"

  • The backpack can weigh at most 25 pounds. → w ≤ 25

• ≤ means "no more than"

  • Spend no more than $50. → c ≤ 50

• ≥ means "no less than"

  • Earn no less than 70 points. → p ≥ 70

Watch Out!

Some phrases can be tricky:

• "More than 5" → x > 5

• "5 more than a number" → x + 5 (this is addition, not an inequality!)

Likewise:

• "Less than 10" → x < 10

• "7 less than a number" → x - 7 (this describes subtraction, not an inequality)

Quick tip: Circle or highlight clue words like at least, more than, or no more than before writing your inequality. These words tell you which symbol to use!

Some phrases may look the same, but they mean completely different things.

• 12 greater than x means 12 more than x, so you add 12:

  • x + 12

• 12 is greater than x compares two values, so it becomes an inequality:

  • 12 > x

When used with greater than or less than, the keyword "is" usually means you're comparing values and not performing an operation.

The sentence is telling you how two values compare. In 12 is greater than x, the number 12 is the larger value.

The phrase 12 greater than x describes an operation. It means start with x, then add 12.

Quick tip: If a phrase has "greater than" or "less than" without the word is, it's often describing an operation. If it says "is greater than" or "is less than," it's comparing two values and should be written as an inequality.

Watch for "Equal To"

Some phrases mean the value can be equal to a number. These use the symbols ≥ or ≤.

• At least means the value is equal to or greater than a number (≥).

• At most means the value is equal to or less than a number (≤).

Quick tip: If you see the words at least or at most, remember to include the equal to part of the inequality.

Examples ✍🏼

At least 12 students must attend the field trip.

• The phrase "at least" means the number of students can be 12 or more.

• Let s = the number of students.

• s ≥ 12

• This means 12 students is acceptable, but any number greater than 12 also works.

The backpack can weigh at most 25 pounds.

• The phrase "at most" means the weight can be 25 pounds or less.

• Let w = the weight of the backpack.

• w ≤ 25

• The backpack can weigh 25 pounds, but it cannot weigh more than 25 pounds.

Rules are important in math because they provide everyone with a specific way to solve a problem.

When solving word problems involving inequalities, you need to remember these two rules!

Rule 1

Inequality symbols will change direction when you swap the left and right-hand sides.

For example, 12 > 2x + 8. If we want the "x" on the left-hand side, we have to reverse the inequality: 2x + 8 < 12.

This makes sense when you consider that the open end of the inequality is facing 12. When we reverse the inequality, we keep the open end pointing to 12.

After solving this, you find that 2 > x or x < 2. Read this: two is greater than x or x is less than two. It says the same thing.

Rule 2

Inequalities will change direction when multiplying or dividing inequalities by negative numbers.

For example: -2y < -8. When you divide both sides by -2, you reverse the inequality. Your answer becomes y > 4 because when we put 4 back into the equation and solve, we see -2(4) < -8 = -8 < -8 is not true.

But choose a number greater than 4, say 5: -2(5) < -8 = -10 < -8...and the inequality is true! So the value of y has to be greater than 4.

Reversing the inequality keeps the equation true!

Todd has $500 in a savings account at the beginning of the summer. He wants to have at least $200 in the account by the end of the summer. He withdraws $25 each week for food, clothes, and movie tickets.

• Write an inequality that represents Todd's situation.

• How many weeks can Todd withdraw money from his account?

How to solve it:

1. Notice keywords that will help you translate the problem.

   • $500 in savings

   • have at least $200 by the end of summer

   • withdraws $25 each week

2. Identify the unknown.

   Let w represent "How many weeks?"

3. Write the inequality.

   Since the phrase at least is used, our translation will use the greater than or equal to symbol.

4. Todd begins with $500 and takes out $25 each week. He wants to keep at least $200 by the end of summer.

5. Solve the inequality.

6. Notice the reverse inequality!

Note: since we divided by -25 after subtracting 500 from both sides, we reversed the inequality's direction.

Use what you've learned to translate inequality word problems and solve them!

If you get stuck or need more help, check out the resources: