9.4 Inequalities on a number line and set notation — Learn

Prerequisites: ordering numbers and reading number lines.

Objectives: interpret and write inequalities, represent their solutions on a number line, and describe solution sets using set-builder and interval notation.

This section covers writing and representing linear inequality solution sets, within AQA reference A22.

An inequality compares values that may differ. The symbols $<$ and $>$ exclude the boundary. The symbols $\leq$ and $\geq$ include it.

Example 1. Show $x<3$ on a number line.

The boundary is $3$, and it is not included, so use an open circle.

Values less than $3$ lie to the left, so shade left.

Example 2. Describe the set $-2\leq x<4$.

The lower boundary $-2$ is included; the upper boundary $4$ is excluded.

The values run from $-2$ up to, but not including, $4$.

For integer values, the set is $\{-2,-1,0,1,2,3\}$. Braces list the members of a set. Set-builder notation describes a rule: $\{x\in\mathbb{R}: -2\leq x<4\}$. Here $\mathbb{R}$ means all real numbers, and the colon means “such that”. Interval notation writes the same range as $[-2,4)$. A square bracket includes an endpoint; a round bracket excludes it.

Watch out. A filled circle means the boundary is included. Use it for $\leq$ or $\geq$; an open circle marks $<$ or $>$.

Predict: before moving either boundary, say which direction represents $x>2$. Test: set the boundary to $2$ and choose the right-facing solution. Explain: why is the circle open, and what change would make it filled?

You should now be able to

• translate phrases such as “at least” and “less than” into symbols;
• draw strict and inclusive inequalities on a number line;
• list integer values and write solution sets in set-builder or interval notation.