Algebra
Linear Inequalities
Year 7 · Year 8 · Year 9
- ✓By the end of this lesson students will be able to understand and use inequality symbols.
- ✓By the end of this lesson students will be able to solve linear inequalities using inverse operations.
- ✓By the end of this lesson students will be able to represent the solution to a linear inequality on a number line.
- ✓By the end of this lesson students will be able to solve linear inequalities involving negative coefficients, remembering to reverse the inequality sign.
Key concepts
An inequality is a mathematical statement that compares two expressions using an inequality symbol, showing that one expression is not necessarily equal to the other. Unlike equations which use an equals sign (=) and usually have a single solution, inequalities use symbols like <, >, ≤, or ≥ and often have a range of solutions.
There are four main inequality symbols: * < means 'less than' (e.g., x < 5 means x can be any number smaller than 5, but not 5 itself). * > means 'greater than' (e.g., x > 2 means x can be any number larger than 2, but not 2 itself). * ≤ means 'less than or equal to' (e.g., x ≤ 7 means x can be any number smaller than or equal to 7, including 7). * ≥ means 'greater than or equal to' (e.g., x ≥ -1 means x can be any number larger than or equal to -1, including -1).
Solving linear inequalities is very similar to solving linear equations. You use inverse operations (addition, subtraction, multiplication, division) to isolate the variable. The key difference is that if you multiply or divide both sides of the inequality by a negative number, you must reverse the direction of the inequality sign.
Solutions to inequalities can be clearly shown on a number line: * Use an open circle (○) for strict inequalities (< or >). This indicates that the number itself is not included in the solution. * Use a closed circle (●) for inclusive inequalities (≤ or ≥). This indicates that the number itself is included in the solution. * Draw an arrow from the circle in the direction of the solution set. For 'less than' or 'less than or equal to', the arrow points left. For 'greater than' or 'greater than or equal to', the arrow points right.
Key facts to remember
- 1Inequalities use symbols: < (less than), > (greater than), ≤ (less than or equal to), ≥ (greater than or equal to).
- 2Solving inequalities is similar to solving equations, using inverse operations.
- 3Crucial Rule: If you multiply or divide both sides of an inequality by a negative number, you MUST reverse the direction of the inequality sign.
- 4On a number line, an open circle (○) means the number is NOT included in the solution (for < or >).
- 5On a number line, a closed circle (●) means the number IS included in the solution (for ≤ or ≥).
- 6The arrow on the number line points left for 'less than' solutions and right for 'greater than' solutions.
Worked examples
Example 1
Solve the inequality 3x - 4 < 11 and represent the solution on a number line.
Answer
x < 5
Remember to use an open circle for 'less than' or 'greater than'.
Example 2
Solve the inequality 7 - 2x ≥ 13 and represent the solution on a number line.
Answer
x ≤ -3
This example highlights the crucial rule: reverse the inequality sign when multiplying or dividing by a negative number.
Example 3
Solve the inequality 5(x + 1) > 2x - 1 and represent the solution on a number line.
Answer
x > -2
Always simplify both sides of the inequality before isolating the variable.
Common mistakes
- ✗Forgetting to reverse the inequality sign when multiplying or dividing by a negative number.
- ✗Confusing open (○) and closed (●) circles on the number line, especially for strict vs. inclusive inequalities.
- ✗Incorrectly drawing the direction of the arrow on the number line (e.g., drawing left for x > 5).
- ✗Making arithmetic errors when adding, subtracting, multiplying, or dividing, just as in equations.
Exam tips
- ★Always double-check your work, especially when you've multiplied or divided by a negative number – this is where most mistakes happen.
- ★Draw your number lines clearly and label key points, including the value where the circle is placed.
- ★To check your answer, pick a number within your solution range and substitute it back into the original inequality. It should make the inequality true. Then pick a number outside your range; it should make it false.
- ★Read the question carefully to ensure you use the correct inequality symbol and represent it accurately on the number line.
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