Solving Inequalities Quizzes

Introduction

Solving inequalities is a fundamental skill in GCSE Maths. Inequalities are mathematical expressions that show the relationship between two quantities that are not equal. Instead of an equals sign, inequalities use symbols such as <, >, ≤, and ≥ to compare values. Mastering inequalities is essential for solving problems in algebra, graphing, and real-life contexts such as budgeting, speed limits, or temperature ranges.

For example, the inequality $$2x + 5 < 13$$ can be solved to find the range of values for x that make the statement true. Understanding how to solve inequalities allows students to determine all possible solutions and represent them on a number line or in interval notation.

Core Concepts

Inequality Symbols

• < : Less than
• > : Greater than
• ≤ : Less than or equal to
• ≥ : Greater than or equal to

Solving Simple Inequalities

The principles for solving inequalities are similar to solving linear equations:

• Use inverse operations to isolate the variable
• Perform the same operation on both sides

Example:

• $$x + 5 < 12$$ → subtract 5: $$x < 7$$
• $$3x > 9$$ → divide by 3: $$x > 3$$

Multiplying or Dividing by Negative Numbers

When multiplying or dividing both sides of an inequality by a negative number, the inequality symbol reverses:

• Example: $$-2x < 6$$ → divide by -2 → $$x > -3$$
• Example: $$-3x + 5 ≥ 2$$ → subtract 5: $$-3x ≥ -3$$ → divide by -3 → $$x ≤ 1$$

Compound Inequalities

Some inequalities involve two comparisons at once, also called double inequalities:

Example:

• $$2 < x + 3 ≤ 7$$
• Subtract 3 from all parts: $$-1 < x ≤ 4$$

Graphical Representation

Solutions of inequalities can be represented on a number line:

• Open circle for < or > (not including the number)
• Closed circle for ≤ or ≥ (including the number)
• Shade left for < or ≤, shade right for > or ≥

Solving Inequalities with Brackets

Expand brackets first, then solve:

Example:

• $$2(x + 3) > 10$$
• Step 1: Expand: $$2x + 6 > 10$$
• Step 2: Subtract 6: $$2x > 4$$
• Step 3: Divide by 2: $$x > 2$$

Inequalities with Fractions

When inequalities involve fractions, multiply both sides by the denominator (considering sign) to solve:

Example:

• $$\frac{x}{3} + 2 ≤ 5$$ → subtract 2: $$\frac{x}{3} ≤ 3$$ → multiply by 3: $$x ≤ 9$$

Checking Solutions

Substitute a value from the solution set into the original inequality to verify:

Example:

• $$x + 5 < 12$$, solution $$x < 7$$
• Test: x = 6 → 6 + 5 = 11 < 12 ✔
• Test: x = 7 → 7 + 5 = 12 ❌ (not part of solution)

Real-Life Applications

• Budgeting: $$x + 20 ≤ 100$$ → money spent plus known expenses
• Speed limits: $$v ≤ 60$$ → speed must be less than or equal to 60 km/h
• Temperature ranges: $$-5 ≤ T ≤ 10$$ → suitable temperature for storage
• Probability: $$0 ≤ p ≤ 1$$ → probability of events
• Time or distance restrictions in real-world contexts

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Worked Examples

Example 1 (Foundation): One-step inequality

$$x + 7 < 12$$

• Subtract 7: $$x < 5$$

Example 2 (Foundation): Multiplication

$$3x ≥ 9$$

• Divide by 3: $$x ≥ 3$$

Example 3 (Higher): Negative coefficient

$$-2x + 5 ≤ 1$$

• Subtract 5: $$-2x ≤ -4$$
• Divide by -2 (reverse inequality): $$x ≥ 2$$

Example 4 (Higher): Brackets

$$2(x + 3) > 10$$

• Expand: $$2x + 6 > 10$$
• Subtract 6: $$2x > 4$$
• Divide by 2: $$x > 2$$

Example 5 (Higher): Fractions

$$\frac{x}{4} + 3 < 7$$

• Subtract 3: $$\frac{x}{4} < 4$$
• Multiply by 4: $$x < 16$$

Example 6 (Higher): Variables both sides

$$3x + 5 > 2x + 7$$

• Subtract 2x: $$x + 5 > 7$$
• Subtract 5: $$x > 2$$

Example 7 (Higher): Double inequality

$$1 < 2x + 3 ≤ 9$$

• Subtract 3: $$-2 < 2x ≤ 6$$
• Divide by 2: $$-1 < x ≤ 3$$

Example 8 (Real-life): Money problem

You can spend up to £50. An item costs £x and shipping is £5. Find inequality:

• $$x + 5 ≤ 50$$
• Subtract 5: $$x ≤ 45$$
• Maximum item price: £45

Example 9 (Real-life): Temperature range

Safe storage: -10°C to 5°C. Let T = temperature. Express as inequality:

• $$-10 ≤ T ≤ 5$$

Example 10 (Higher): Negative and fractions combined

$$-\frac{1}{2}x + 3 ≥ 5$$

• Subtract 3: $$-\frac{1}{2}x ≥ 2$$
• Divide by -1/2 (reverse inequality): $$x ≤ -4$$

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Common Mistakes

• Failing to reverse the inequality when multiplying/dividing by negative
• Forgetting to subtract or add on both sides
• Not expanding brackets before solving
• Confusing ≤, ≥, <, > symbols
• Incorrect handling of fractions

Tips to avoid errors:

• Always perform the same operation on both sides
• Remember to reverse the inequality when multiplying/dividing by a negative number
• Check solution by substituting values
• Use number lines to visualise the solution
• Practice real-life word problems involving inequalities

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Applications

• Budgeting and finance: spending limits
• Physics: speed, temperature, or force ranges
• Probability: ensuring values are within valid ranges
• Algebra: solving equations and inequalities
• Exam problem-solving: interpreting and representing solutions on number lines

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Strategies & Tips

• Isolate the variable systematically
• Reverse inequality when multiplying/dividing by negative
• Expand brackets first if needed
• Visualise solutions on a number line
• Check with substitution for accuracy

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Summary / Call-to-Action

Solving inequalities is a critical skill for GCSE Maths. By mastering one-step, two-step, brackets, fractions, negatives, and real-life inequalities, students can determine solution ranges confidently. Regular practice ensures understanding, accuracy, and readiness for exam questions.

Next Steps:

• Attempt quizzes on inequalities to reinforce learning
• Practice multi-step and compound inequalities
• Apply inequalities in real-life contexts such as budgets, temperature, and speed limits
• Challenge yourself with higher-level problems combining fractions, negatives, and multiple operations

Consistent practice will make solving inequalities intuitive and error-free in all GCSE Maths problems.