Inequalities Word Problems — Step-by-Step Solutions

By MathPal Editorial Team·Published Apr 21, 2026

Grade: 7-8 | Topic: Algebra

What You Will Learn

Word problems often ask for ranges of values rather than single answers — "at least," "no more than," "fewer than." These translate into inequalities. In this guide you will learn to identify key phrases, write an inequality, solve it, and interpret the answer in context.

Theory

Key phrases and their symbols

Phrase	Symbol	Example
is greater than	$>$	more than 10: $x > 10$
is less than	$<$	fewer than 5: $x < 5$
is at least / no less than	$\geq$	at least 18: $x \geq 18$
is at most / no more than	$\leq$	maximum of 100: $x \leq 100$

Solving inequalities

Solve an inequality exactly like an equation — with one critical exception:

When you multiply or divide both sides by a negative number, reverse the inequality sign.

$-3x < 12 \implies x > -4 \quad \text{(sign flipped)}$

Writing the solution

• As an inequality: $x \geq 3$
• On a number line: closed dot at 3, arrow pointing right
• In words: "x can be 3 or any value greater than 3"

Worked Examples

Example 1 — Minimum earnings

Maria earns $12 per hour. She needs to earn at least $180. How many hours must she work?

Step 1: Let $h$ = number of hours. "At least $180" means $\geq 180$:

$12h \geq 180$

Step 2: Divide both sides by 12:

$h \geq 15$

Answer: Maria must work at least 15 hours.

Check: $12 \times 15 = 180$ ✓ (equals 180, which satisfies $\geq 180$). $12 \times 14 = 168 < 180$ ✗ (14 hours is not enough).

Example 2 — Maximum budget

A club has a budget of $250 for supplies. They have already spent $85. How much more can they spend?

Step 1: Let $s$ = additional spending. "No more than $250":

$85 + s \leq 250$

Step 2: Subtract 85:

$s \leq 165$

Answer: The club can spend at most $165 more.

Example 3 — Flipping the sign

A student needs a final exam score so that the average of three tests (82, 76, and the final) is above 80. What score must the student earn?

Step 1: Let $f$ = final score. Average $> 80$:

$\frac{82 + 76 + f}{3} > 80$

Step 2: Multiply both sides by 3:

$158 + f > 240$

Step 3: Subtract 158:

$f > 82$

Answer: The student must score more than 82 on the final.

Example 4 — Negative coefficient (sign flip)

A submarine descends below sea level. Its depth $d$ (negative number) satisfies $-4d \geq 48$. What are the possible values of $d$?

Step 1: Divide both sides by $-4$ — sign flips:

$d \leq -12$

Answer: The depth is -12 or more negative (i.e., at least 12 m below sea level).

Common Mistakes

Mistake 1 — Forgetting to flip the sign

❌ $-5x > 20 \Rightarrow x > -4$

✅ Dividing by $-5$ flips the sign: $x < -4$.

Mistake 2 — Misreading "at least" vs "more than"

❌ "At least 3" written as $x > 3$ (strict inequality).

✅ "At least" includes the boundary: $x \geq 3$. "More than 3" excludes it: $x > 3$.

Mistake 3 — Not checking the answer in context

❌ Getting $h \geq 15.3$ hours and stating the answer as "at least 15.3 hours" when hours must be whole numbers.

✅ Round up to the nearest whole number in context: the student must work at least 16 full hours.

Practice Problems

Problem 1: A box can hold at most 30 kg. It already has 12 kg inside. How much more weight can be added?

Show Answer

$12 + w \leq 30 \Rightarrow w \leq 18$ kg.

At most 18 kg more.

Problem 2: Jake wants to save more than $500. He already has $180 and saves $40 per week. How many weeks until he reaches his goal?

Show Answer

$180 + 40w > 500 \Rightarrow 40w > 320 \Rightarrow w > 8$.

More than 8 weeks — so at least 9 weeks.

Problem 3: Solve and interpret: $-3n + 9 > 21$.

Show Answer

$-3n > 12 \Rightarrow n < -4$ (sign flips when dividing by $-3$).

$n$ must be less than $-4$.

Problem 4: A phone plan allows at most 2 GB of data per day. Over 5 days, Max used 1.2, 0.8, 1.5, 2.0 and $d$ GB. What are the possible values of $d$ if the total must not exceed 10 GB?

Show Answer

$1.2 + 0.8 + 1.5 + 2.0 + d \leq 10 \Rightarrow 5.5 + d \leq 10 \Rightarrow d \leq 4.5$.

But the daily limit also applies: $d \leq 2$.

So $d \leq 2$ (the daily cap is the binding constraint).

Problem 5: Write an inequality: "A rollercoaster requires riders to be at least 120 cm tall. Kenji is 117 cm. How many cm does he need to grow?"

Show Answer

Let $g$ = cm to grow. $117 + g \geq 120 \Rightarrow g \geq 3$.

Kenji needs to grow at least 3 cm.

Summary

• Convert word phrases to symbols: "at least" → $\geq$, "at most" → $\leq$, "more than" → $>$, "less than" → $<$.
• Solve inequalities like equations — but flip the sign when multiplying or dividing by a negative number.
• Check your answer by substituting a value from the solution range into the original inequality.
• In real-world problems, consider whether the answer needs to be a whole number.

Related Topics

• How to Solve Inequalities Step by Step — algebraic techniques without the word-problem context
• Writing and Solving Equations from Word Problems — the equation version of the same skills
• Solving Two-Step Equations — foundational equation solving

---

Need help with inequality word problems? Take a photo of your math problem and MathPal will solve it step by step. Open MathPal