Solving & Graphing InequalitiesCambridge IGCSE Maths: Revision notes

An inequality compares two expressions using $<$, $>$, $\leq$ or $\geq$ instead of $=$.

• $<$ means "less than"; $>$ means "greater than" — these are strict inequalities
• $\leq$ means "less than or equal to"; $\geq$ means "greater than or equal to" — these are inclusive

On a number line:

• An open circle shows a strict inequality ($<$ or $>$) — the boundary value is not included
• A closed (filled) circle shows an inclusive inequality ($\leq$ or $\geq$) — the boundary value is included

Since diagrams cannot be drawn, describe the number-line representation in words: state the boundary value(s), whether each circle is open or closed, and the direction of shading.

Linear inequalities are solved using the same steps as linear equations — with one crucial exception.

Rule: if you multiply or divide both sides by a negative number, the inequality sign must reverse.

1. Collect terms as you would for an equation
2. If a negative multiplication/division is needed, flip the inequality sign

Example: Solve $3x < 2x + 4$. $x < 4$ (no negative operation needed).

Example (sign flip): Solve $-2x > 8$. Dividing both sides by $-2$: $x < -4$ (sign flipped from $>$ to $<$).

Example (double inequality, Extended): Solve $-3 \leq 3x - 2 < 7$. Add 2 throughout: $-1 \leq 3x < 9$. Divide by 3: $-\frac{1}{3} \leq x < 3$.

A linear inequality in two variables (e.g. $y \leq 2x + 3$) can be represented on coordinate axes, but since this platform avoids diagrams, describe the region algebraically instead:

• The boundary line is $y=2x+3$
• A broken (dashed) line is used for strict inequalities ($<$, $>$); a solid line is used for inclusive inequalities ($\leq$, $\geq$)
• Shading convention: the unwanted region is shaded, leaving the required region unshaded — always check which convention a question specifies

To check whether a point satisfies an inequality, substitute its coordinates into the inequality and see if the statement is true.

Example: Does $(1, 5)$ satisfy $y \leq 2x+3$? Substituting: $5 \leq 2(1)+3=5$. Since $5 \leq 5$ is true, yes.

Given a description of a region (or a set of boundary lines with a stated point inside the region), you can list the inequalities that define it:

1. Identify the equation of each boundary line
2. For each line, test a known point inside the region to determine whether the required side is $<$, $>$, $\leq$ or $\geq$
3. Write out all inequalities together — a point lies in the region only if it satisfies all of them simultaneously

Example: A region is bounded by $x \geq 1$, $y \geq 0$, and $x+y \leq 6$. The point $(2,2)$ satisfies all three, so it lies inside the region.