How to Solve Inequality Word Problems
Translate and solve inequality word problems correctly, including the negative-multiplication sign flip, with examples and practice questions.
Expected Interview Answer
Inequality word problems are solved by translating phrases like “at least,” "at most," "more than," and “fewer than” into <, ≤, >, or ≥ statements, then solving them like equations except that multiplying or dividing by a negative number flips the inequality sign.
The translation step is the highest-risk part: "at least 5" means ≥ 5 (5 is included), while “more than 5” means > 5 (5 is excluded), and mixing these up is the single most common error. Once translated, solving proceeds exactly like a linear equation — isolate the variable through valid operations — with the one critical exception that multiplying or dividing both sides by a negative number reverses the inequality direction. For compound conditions joined by “and,” intersect the solution ranges; for “or,” take the union. The final answer should always be expressed as a range or interval, and sanity-checked against the real-world constraint, such as a quantity that cannot be negative.
- Precise phrase-to-symbol translation avoids the most common scoring error
- The flip-on-negative-multiplication rule is the one exception to treat like an equation
- Expressing answers as ranges keeps them interview-ready and unambiguous
AI Mentor Explanation
A team needs “at least 180” to win, meaning 180 itself is enough — that is a ≥ boundary, not a strict > one, and confusing the two changes whether the exact target score counts as a win. If the required run rate condition also says the team must score in “fewer than 20 overs,” that combines a ≥ condition on runs with a < condition on overs, and both must hold simultaneously — an intersection, exactly like the “and” rule in inequality word problems. Getting the boundary inclusion wrong is the cricket equivalent of a scorer misjudging a tie versus a win.
Worked example
Inequality
- 95 + x ≥ 240
Isolate x
- x ≥ 240 − 95
Solution
- x ≥ 145
Step-by-Step Explanation
Step 1
Translate the phrase
"At least" → ≥, "at most" → ≤, "more than" → >, "fewer than" → <.
Step 2
Set up the inequality
Write the condition using the correct variable and symbol.
Step 3
Solve like an equation
Isolate the variable through valid arithmetic operations.
Step 4
Flip on negative multiplication
Reverse the inequality sign only when multiplying or dividing by a negative number.
What Interviewer Expects
- Correct translation of boundary phrases into ≤, <, ≥, > symbols
- Correct sign-flip rule when multiplying/dividing by a negative number
- Correct intersection ("and") vs union ("or") handling for compound conditions
- Answer expressed as a clear, real-world-sensible range
Common Mistakes
- Treating “at least” as strict (>) instead of inclusive (≥)
- Forgetting to flip the inequality sign when multiplying/dividing by a negative
- Taking a union when the problem actually requires an intersection
- Not checking that the solution range respects real-world constraints (e.g., non-negative quantities)
Best Answer (HR Friendly)
“I first translate the wording precisely — "at least" and “at most” include the boundary, while “more than” and “fewer than” exclude it — since that single word decides whether the edge value counts. Then I solve the inequality exactly like an equation, with one exception: multiplying or dividing both sides by a negative number flips the direction of the inequality. For compound conditions, I intersect ranges for “and” and take the union for “or,” and I always double-check the final range makes sense in context.”
Follow-up Questions
- How do you solve a compound inequality with a variable on both sides?
- How does solving an inequality differ when the coefficient of x is negative?
- How would you represent the solution of an inequality on a number line?
- How do you handle a word problem combining an equation and an inequality together?
MCQ Practice
1. Solve for x: 3x + 5 ≤ 20.
3x ≤ 15, so x ≤ 5.
2. Solve for x: −2x + 4 > 10.
−2x > 6 → dividing by −2 flips the sign: x < −3.
3. A parking lot charges a flat 20 plus 5 per hour, and a customer wants to pay at most 45. What is the maximum number of whole hours they can park?
20 + 5h ≤ 45 → 5h ≤ 25 → h ≤ 5, so the maximum whole number of hours is 5.
Flash Cards
"At least" translates to? — ≥ (boundary value included).
"More than" translates to? — > (boundary value excluded).
When does an inequality sign flip? — When multiplying or dividing both sides by a negative number.
"And" vs “or” for compound inequalities? — "And" takes the intersection of ranges; "or" takes the union.