Linear Inequalities

Grade 4

Inequalities and number lines

An inequality gives a range of values rather than one answer. The symbols are: < (less than), > (greater than), ≤ (less than or equal to), ≥ (greater than or equal to).

You solve a linear inequality almost identically to a linear equation — with one critical difference.

The one rule that differs from equations

When you multiply or divide both sides of an inequality by a negative number, the inequality sign flips direction. So −2x < 6 becomes x > −3 after dividing by −2. Get this wrong and your answer points the wrong way.

Worked example

EXAMPLE

Solve 3x − 5 ≥ 7 and represent the answer on a number line.

Add 5 to both sides: 3x ≥ 12. Divide by 3 (positive, so sign stays): x ≥ 4. On a number line: closed circle at 4 (because ≥ includes 4), arrow pointing right. x ≥ 4.

Number line notation

On a number line, an open circle (○) means the endpoint is not included (use for < and >). A closed/filled circle (●) means the endpoint is included (use for ≤ and ≥).

Try this one

Solve the inequality

\[ 2x^2 - 5x - 3 \ge 0 \]

[3]

Nova's hint:

This is a quadratic inequality. Factorise the quadratic first to find the critical values where it equals zero.

Ask yourself: 'for an upward parabola with a \( \ge 0 \) sign, do I want the region between the roots or outside them?' A rough sketch of the curve makes the choice clear.

Try to Solve it with Nova?