Inequalities

• Represent inequalities on a number line: open circles for < and >, closed circles for ⩽ and ⩾.
• Construct and solve linear inequalities, including double inequalities such as −3 ⩽ 3x − 2 < 7.
• List integer solutions.
• Represent linear inequalities in two variables on a graph: broken lines for strict, solid lines for inclusive, shade the unwanted region.
• List the inequalities that define a given region. Linear programming is not included.

Open or closed circles

For −3 ⩽ x < 1, draw a closed circle at −3, an open circle at 1 and a line joining them. The integers in this range are −3, −2, −1 and 0.

Like an equation, except for negatives

Add, subtract, multiply or divide both sides as for an equation. If you multiply or divide by a negative number, reverse the inequality sign. For a double inequality, do the same to all three parts.

Boundary line, then shade the side you do not want

Draw the boundary line as if it were an equation: broken for < or >, solid for ⩽ or ⩾. Test a point not on the line (the origin is easiest) to decide which side satisfies the inequality, then shade the unwanted side, unless the question says otherwise. The region left unshaded is the solution.