Tayarri LCAT
LCAT Math, Algebra

Linear inequalities for the LCAT: lesson and practice

Solve an inequality exactly like an equation, with one extra rule: when you multiply or divide both sides by a negative number, flip the sign. In word problems, "at most" means ≤ and "at least" means ≥. For whole-number answers, round in the direction that keeps the inequality true.

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No calculator on the LCAT. LUMS’s official guidelines say calculators are not allowed, so every practice question here can be solved by hand.

The one extra rule

−3x + 7 > 22
−3x > 15
x < −5 (sign flipped when dividing by −3)

Translating words

WordsSymbol
at most, no more than, maximum≤
at least, no less than, minimum≥
more than, exceeds>
fewer than, below<

Example

A plan costs Rs 300 plus Rs 90 per GB. Hira can spend at most Rs 2,000. What is the most whole GB she can use?

300 + 90g ≤ 2,000
90g ≤ 1,700
g ≤ 18.9, so 18 GB

Rounding up to 19 would break her budget.

Common mistakes

Forgetting to flipOnly when multiplying or dividing by a negative.
Boundary errors< and > exclude the boundary; ≤ and ≥ include it.
Reading "at most" as ≥"At most" is a ceiling: ≤.

Sample questions with answers

1. Which describes all solutions to −2x + 4 > 10?
  1. x > −3
  2. x < −3 (correct)
  3. x < 3
  4. x ≤ −3

−2x > 6, and dividing by −2 flips the sign: x < −3.

2. What is the greatest integer x with x/3 − 4 < 2?
  1. 18
  2. 17 (correct)
  3. 6
  4. 5

x/3 < 6, so x < 18. The greatest integer is 17.

3. A lift carries at most 800 kg. A 70 kg operator carries 20 kg boxes. Which inequality fits the number of boxes b?
  1. 70 + 20b ≥ 800
  2. 20 + 70b ≤ 800
  3. 70 + 20b ≤ 800 (correct)
  4. 90b ≤ 800

"At most" means ≤, and each box adds 20 kg to the operator’s 70 kg.

Frequently asked questions

When do you flip the inequality sign?

Only when you multiply or divide both sides by a negative number.

What does "at most" mean in an LCAT inequality?

Less than or equal to (≤). "At least" means greater than or equal to (≥).

How do you solve a compound inequality?

Do the same operation to all three parts at once, then read the range. Check whether each end is included.

How do I find a solution to a system of inequalities?

Test each answer choice in every inequality. A solution must satisfy all of them.

Is linear inequalities tested on the LCAT?

Yes. It falls under Algebra, one of the content domains LUMS lists for the LCAT in its official sample questions.

Can I practice this without a calculator?

Yes. Every practice question here can be solved by hand, because calculators are not allowed on the LCAT.

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