InequalitiesEdexcel GCSE Maths: Revision notes

Inequality symbols are used to show relationships between values that are not equal.

• < and > are strict inequalities (the value is not included)
• ≤ and ≥ are non-strict inequalities (the value is included)
• When writing inequalities, the pointed end always faces the smaller value
• An inequality describes a range of values, not just one number

Solving linear inequalities is almost identical to solving equations, with one crucial difference:

When you multiply or divide both sides by a negative number, you must reverse the inequality symbol.

Step-by-step process:

1. Rearrange the inequality as you would an equation (add, subtract, multiply, divide)
2. If multiplying or dividing by a negative number, flip the inequality symbol
3. Write the final answer clearly

Example 1: Solve 3x + 2 < 11

• 3x < 9
• x < 3

Example 2: Solve -2x + 5 ≥ 13

• -2x ≥ 8
• x ≤ -4 (symbol reversed because we divided by -2)

Key difference from equations:

• 2x = 6 gives x = 3 (one value)
• 2x < 6 gives x < 3 (many values)

Number line representation:

• Use an open circle (○) for strict inequalities (< or >)—the boundary is not included
• Use a closed circle (●) for non-strict inequalities (≤ or ≥)—the boundary is included
• Draw an arrow along the line in the direction the inequality points

Example: x ≤ 4 is shown with a closed circle at 4 and an arrow pointing left (towards smaller values).

Set notation:

• Write solutions using set-builder notation or interval notation
• Set-builder notation: {x : x < 3} means "the set of all x such that x is less than 3"
• Can also write {x ∈ ℝ : x ≤ 5} to specify x is a real number

Common forms:

• {x : x > -2} represents all values greater than -2
• {x : 1 ≤ x < 6} represents values from 1 (included) to 6 (not included)
• {x : x ≥ 10} represents 10 and all values above it

Solving quadratic inequalities requires finding where a quadratic expression is positive or negative.

Method:

1. Rearrange to standard form: e.g. x² - 5x + 6 > 0
2. Factorise the quadratic (or use the quadratic formula if needed): (x - 2)(x - 3) > 0
3. Find the roots by solving (x - 2)(x - 3) = 0, giving x = 2 and x = 3
4. Sketch or analyse a sign diagram to determine where the expression is positive or negative
5. Write the solution using inequality notation

Sign diagram analysis:

• For (x - 2)(x - 3) > 0, test regions:
  • x < 2: both factors negative → product positive ✓
  • 2 < x < 3: one positive, one negative → product negative ✗
  • x > 3: both factors positive → product positive ✓
• Solution: x < 2 or x > 3

Example: Solve x² - 5x + 4 ≤ 0

• Factorise: (x - 1)(x - 4) ≤ 0
• Roots: x = 1, x = 4
• Between roots the product is negative: 1 ≤ x ≤ 4

Key point: Whether the inequality includes equals (≤, ≥) affects whether boundary values are in the solution.

Linear inequalities in two variables:

An inequality like y < 2x + 1 describes a region of the coordinate plane, not just a line.

Steps:

1. Draw the boundary line y = 2x + 1
   • Use a dashed line for < or > (boundary not included)
   • Use a solid line for ≤ or ≥ (boundary included)
2. Test a point not on the line (usually the origin (0,0)) to determine which side satisfies the inequality
3. Shade the region that satisfies the inequality

Quadratic inequalities in two variables:

For y ≤ x², the boundary is a parabola:

1. Draw the parabola y = x² using a solid line (non-strict inequality)
2. Test a point (e.g. (0, 1)) to determine the region
3. Shade the region below the parabola (where y ≤ x²)

Multiple inequalities:

When solving a system of inequalities (e.g. y < 2x + 1 AND y ≥ x - 2), the solution is the intersection of all regions—where all conditions are satisfied simultaneously. Shade each region lightly first, then the overlapping region darker or clearly identify it.

Key points:

• Dashed lines (boundary excluded) vs solid lines (boundary included) matter
• The shading shows all points (x, y) that satisfy the inequality
• Always test a point to confirm which side you're shading