Linear Inequalities

Use inverse operations exactly as for equations, keeping the inequality sign in place of the equals sign.

If you multiply or divide both sides by a negative number, $\text{reverse the sign}$: $<$ becomes $>$ and $\le$ becomes $\ge$.

You can often avoid this by collecting the unknown on the side where its coefficient is positive.

On a number line, an $\text{open circle}$ means the endpoint is not included ($<$ or $>$); a $\text{filled circle}$ means it is included ($\le$ or $\ge$). An arrow or line shows the rest of the solution.

A solution such as $-2<x\le 5$ is drawn as a line segment from an open circle at $-2$ to a filled circle at $5$.

In set notation, $\{x:x>3\}$ means "the set of all $x$ such that $x>3$".

For a double inequality like $-4<3n+2\le 11$, apply each step to all three parts.

Integer solutions are the whole numbers in the solution set — watch whether the endpoints are included.

An inequality in $x$ and $y$ describes a region. Draw the boundary line as if it were an equation: a $\text{solid}$ line for $\le$ or $\ge$ (included), a $\text{dashed}$ line for $<$ or $>$ (not included).

Test a point not on the line, such as the origin, to decide which side satisfies the inequality.

For several inequalities, the required region is where all are satisfied. Edexcel usually asks you to shade it and label it $R$.