Solving Inequalities Quizzes
Visual overview of Solving Inequalities.
Introduction
Solving inequalities means finding all possible values of a variable that make an inequality true. Instead of an equals sign, inequalities use the symbols <, >, ≤, and ≥ to compare quantities. They are used in GCSE Maths to describe ranges of solutions and appear in algebra, graphing, and real-life problems such as budgeting, temperature, or speed limits.
Example: \(2x+5<13 \Rightarrow x<4\).
Core Concepts
Inequality Symbols
- < – less than
- > – greater than
- ≤ – less than or equal to
- ≥ – greater than or equal to
Solving Simple Inequalities
The process is almost identical to solving linear equations:
- Use inverse operations to isolate the variable.
- Do the same operation to both sides to maintain balance.
Examples
- \(x+5<12\Rightarrow x<7\)
- \(3x>9\Rightarrow x>3\)
Multiplying or Dividing by Negatives
When multiplying or dividing both sides by a negative number, reverse the inequality sign.
- \(-2x<6\Rightarrow x>-3\)
- \(-3x+5≥2\Rightarrow -3x≥-3\Rightarrow x≤1\)
Compound (Double) Inequalities
A compound inequality contains two comparisons at once:
- \(2<x+3≤7\)
- Subtract 3 from all parts: \(-1<x≤4\)
Graphical Representation
- Open circle – for < or > (number not included).
- Closed circle – for ≤ or ≥ (number included).
- Shade left for < or ≤, shade right for > or ≥.
Inequalities with Brackets
Expand brackets first, then solve normally.
- \(2(x+3)>10\Rightarrow 2x+6>10\Rightarrow x>2\)
Inequalities with Fractions
Multiply both sides by the denominator (taking care with negatives).
- \(\frac{x}{3}+2≤5\Rightarrow \frac{x}{3}≤3\Rightarrow x≤9\)
Checking Solutions
Substitute a value from your solution set to test.
- For \(x+5<12\), solution \(x<7\).
- Test \(x=6\Rightarrow 6+5=11<12\) ✓; \(x=7\Rightarrow12<12\) ✗
Real-Life Contexts
- Budget: \(x+20≤100\) → money limit.
- Speed limit: \(v≤60\).
- Temperature range: \(-5≤T≤10\).
- Probability: \(0≤p≤1\).
Worked Examples
Example 1 (Foundation): One-step
\(x+7<12\Rightarrow x<5\)
Example 2 (Foundation): Multiplication
\(3x≥9\Rightarrow x≥3\)
Example 3 (Higher): Negative coefficient
\(-2x+5≤1\Rightarrow -2x≤-4\Rightarrow x≥2\)
Example 4 (Higher): Brackets
\(2(x+3)>10\Rightarrow x>2\)
Example 5 (Higher): Fractions
\(\frac{x}{4}+3<7\Rightarrow x<16\)
Example 6 (Higher): Variables both sides
\(3x+5>2x+7\Rightarrow x>2\)
Example 7 (Higher): Double inequality
\(1<2x+3≤9\Rightarrow -1<x≤3\)
Example 8 (Real-life): Budget
You can spend up to £50: \(x+5≤50\Rightarrow x≤45\)
Example 9 (Real-life): Temperature range
Safe storage: \(-10≤T≤5\).
Example 10 (Higher): Negatives and fractions
\(-\frac{1}{2}x+3≥5\Rightarrow -\frac{1}{2}x≥2\Rightarrow x≤-4\)
Common Mistakes
- Not reversing the inequality when dividing/multiplying by negatives.
- Missing steps or forgetting to act on both sides.
- Forgetting to expand brackets first.
- Mixing ≤, ≥, <, > symbols.
- Mishandling fractions.
Applications
- Budgeting and finance (limits and constraints).
- Physics: speed, temperature, and force ranges.
- Probability and statistics: values between 0 and 1.
- Algebra and graphing inequalities.
Strategies & Tips
- Isolate the variable step by step.
- Reverse the symbol if multiplying/dividing by a negative.
- Expand brackets before solving.
- Check solutions using substitution.
- Use number lines to visualise solution sets.
Summary / Call-to-Action
Inequalities describe ranges of possible solutions instead of single answers. By learning to isolate variables, handle negatives, and represent solutions on number lines, you can solve inequalities confidently in exams and real life.
- Practise one- and two-step inequalities daily.
- Draw number lines for clear visual understanding.
- Mix brackets, fractions, and negatives for advanced practice.
- Apply inequalities in budgeting and science contexts.