Linear inequalities

An inequality describes a set of values

For 3x < 12, divide by positive 3 to obtain x < 4. The answer includes every real value below 4 unless the question supplies a narrower domain. On a number line, an open circle excludes the endpoint; a filled circle includes it for ≤ or ≥.

Negative multiplication reverses order

Although 2 < 5, multiplying both by −1 gives -2 > -5. The order reverses on the number line. Therefore -4x ≤ 12 gives x ≥ -3, not x ≤ -3.

Adding or subtracting the same number on both sides preserves order. For 5-2x > 11, subtract 5: -2x > 6. Divide by −2 and reverse: x < -3. Check an included value, such as −4, and an excluded value, such as 0.

Check your understanding 1 Use the stated number set

Solve -3x ≥ 15. List all integer solutions with -7 ≤ x ≤ 2.

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The inequality gives x ≤ -5. In the given integer interval the solutions are −7, −6 and −5. The endpoint −5 is included because the original relation is ≥.

Equality is a boundary, not the whole answer

A learner solves 2x + 1 < 9 by writing x = 4. Correct the conclusion and explain what 4 represents.