Inequalities

Spec reference: Algebra - Solving equations and inequalities
Key idea: Solve and represent linear inequalities on a number line and on a graph.

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Inequality symbols

Symbol	Meaning
$>$	Greater than
$<$	Less than
$\ge$	Greater than or equal to
$\le$	Less than or equal to

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Number line representation

• Open circle $\circ$ for strict inequalities ($>$ or $<$)
• Closed circle $\bullet$ for inclusive inequalities ($\ge$ or $\le$)

Example

Show $x>2$ on a number line.

Open circle at 2, arrow pointing right.

Example

Show $-1\le x<4$ on a number line.

Closed circle at $-1$, open circle at $4$, line between them.

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Solving linear inequalities

Solve like an equation, but if you multiply or divide by a negative number, flip the inequality sign.

Example

Solve $3x+2>11$

$3x>9$

$$\boxed{{\displaystyle x>3}}$$

Example

Solve $-2x\le 8$

Divide by $-2$ and flip the sign:

$$\boxed{{\displaystyle x\ge -4}}$$

Example

Solve $2\le 3x-1<14$

Add 1 throughout: $3\le 3x<15$

Divide by 3: $\boxed{{\displaystyle 1\le x<5}}$

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Integer solutions

Example

List the integers that satisfy $-2<x\le 3$

The integers are: $\boxed{{\displaystyle -1,0,1,2,3}}$

(Note: $-2$ is excluded because the inequality is strict, but $3$ is included.)

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Graphical inequalities

To show a region on a graph:

1. Draw the boundary line (dashed for $>$ or $<$, solid for $\ge$ or $\le$)
2. Shade the required region
3. Test a point to check you have the correct side

Example

Show the region $y>2x-1$

Draw the line $y=2x-1$ as a dashed line.

Test $(0,0)$: $0>-1$ ✓. Shade the side containing $(0,0)$.

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Exam tips

Watch out for

• Flip the inequality sign when multiplying or dividing by a negative
• Open circle = value NOT included; closed circle = value IS included
• For graphical inequalities, always test a point to confirm which side to shade

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Test Yourself

Question 1 of 5

Solve 3x+2>11