Linear Equations & InequalitiesCambridge IGCSE Maths: Revision notes

A linear equation contains the unknown only to the power 1. Solve by performing the same operation to both sides until the unknown is isolated.

Example: $3x + 4 = 10$

1. Subtract 4: $3x = 6$
2. Divide by 3: $x = 2$

When brackets and unknowns appear on both sides, expand first, then collect all unknown terms on one side:

$5 - 2x = 3(x+7)$

1. Expand: $5 - 2x = 3x + 21$
2. Add $2x$ to both sides: $5 = 5x + 21$
3. Subtract 21: $-16 = 5x$
4. Divide: $x = -3.2$

Inequalities describe a range of values rather than one exact value.

• Use an open circle for strict inequalities ($<$, $>$) — the boundary value is not included
• Use a closed circle for inclusive inequalities ($\leq$, $\geq$) — the boundary value is included
• Shade or draw an arrow in the direction of all included values

For example, $x > 2$ is shown with an open circle at 2 and an arrow extending to the right.

Solve inequalities the same way as equations, EXCEPT: multiplying or dividing both sides by a negative number reverses the inequality sign.

Example: $3x < 2x + 4$

1. Subtract $2x$: $x < 4$

Compound inequalities can be solved in one line, applying the same operation to all three parts:

$-3 \leq 3x - 2 < 7$

1. Add 2 to all parts: $-1 \leq 3x < 9$
2. Divide all parts by 3: $-\frac{1}{3} \leq x < 3$

To represent an inequality like $y \leq 2x + 1$ graphically:

1. Draw the boundary line $y = 2x + 1$: use a broken (dashed) line for strict inequalities ($<$, $>$), and a solid line for inclusive inequalities ($\leq$, $\geq$)
2. Shade the unwanted region (the side that does NOT satisfy the inequality), leaving the required region clear
3. To list the inequalities defining a shaded region, identify each boundary line's equation and the correct inequality direction for the unshaded side