Linear inequalities in one or two variables

Inequalities solve almost exactly like equations, with one rule that trips people up: flip the inequality sign whenever you multiply or divide by a negative number.

Tested on SAT Algebra

What College Board tests

Solving one-variable inequalities, translating "at most / at least" word problems into inequalities, and checking whether a point satisfies an inequality or a system of inequalities (a shaded region in the xy-plane).

The one rule that's different

Treat an inequality like an equation — add, subtract, multiply, divide both sides — with a single exception: when you multiply or divide both sides by a negative number, reverse the inequality sign.

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Four worked examples in SAT format. Read the approach, try it yourself, then tap Show the full solution.

1 · Solve an inequality (watch the flip)

Which of the following describes all solutions to ?

Approach Isolate just like an equation. The last step divides by a negative number — that's the moment to flip the sign.

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Answer: B

the given inequality

subtract 5 from both sides

divide by −3 — dividing by a negative flips < to >

Why the other choices are wrong

A  Correct number, but forgets to flip the sign.

C  Divides incorrectly and doesn't flip.

D  Drops the negative on the 5 in the result.

2 · Translate "at most" into an inequality

A delivery van starts a trip with a $50 base cost and adds $12 for each package delivered. The total cost cannot exceed $200. If is the number of packages, what is the greatest number of packages that can be delivered?

Approach "Cannot exceed $200" means the total is . Build the cost expression, solve the inequality, then remember packages must be a whole number — round down to stay under the limit.

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Answer: B

total cost is at most 200

subtract 50 from both sides

divide both sides by 12 (positive — no flip)

packages are whole; round down to stay at or under the limit

The $12 charge is a rate. "$12 per package" means "$12 for each one package," so its denominator is 1 package:

For 12 packages the "package" unit cancels, leaving dollars:

Adding the $50 base gives $194 — at or under $200. A 13th package would add another $12, pushing the total to $206, over the limit.

Why the other choices are wrong

A  Rounds down too far — 12 packages still fits.

C  Rounds 12.5 up, which would exceed $200.

D  Ignores the $50 base cost.

3 · Does a point satisfy the inequality?

Which of the following points is a solution to the inequality ?

Approach A point is a solution if plugging in its and makes the inequality true. Test a point by substituting both coordinates and checking.

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Answer: C

test (0, 3): put ,

3 is greater than −1, so the point satisfies the inequality

Checking the others: gives (false); gives (false); gives (false — it's on the line, not above it).

Why the other choices are wrong

A   is false.

B   is false.

D   is false; strict excludes points on the line.

4 · A point in a system of inequalities

Which point satisfies both inequalities below?

Approach A point in a system must make every inequality true. Test a candidate in both; if either fails, it's out.

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Answer: B

test (1, 2) in the first: — true

test (1, 2) in the second: — true

satisfies both, so it's the solution

Why the other choices are wrong

A  Fails the second: is false.

C  Fails the first: is false.

D  Fails the first: is true, but it fails the second since is false.

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