Inequalities

Inequalities tell us about a range of values, rather than one specific solution. This lesson covers solving linear inequalities, representing solutions on number lines and using set notation, graphical inequalities (Higher), and quadratic inequalities (Higher).

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Key Vocabulary

Symbol	Meaning
<	Less than
>	Greater than
$\leq$≤	Less than or equal to
$\geq$≥	Greater than or equal to

Term	Meaning
Inequality	A mathematical statement comparing two expressions using <, >, $\leq$≤, or $\geq$≥
Integer values	Whole numbers (positive, negative, or zero)
Set notation	e.g. {x : x > 3} meaning "the set of all x such that x is greater than 3"

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Solving Linear Inequalities

Solve exactly like a linear equation — with one important rule: if you multiply or divide by a negative number, you must reverse the inequality sign.

Worked Example 1

Solve: 3x + 5 > 14

3x > 9

x > 3

Worked Example 2

Solve: $7 - 2x \leq 3$7−2x≤3

$-2x \leq -4$−2x≤−4

Divide by −2 (reverse the sign!): $x \geq 2$x≥2

Worked Example 3

Solve: 4(x − 1) < 2x + 10

4x − 4 < 2x + 10

2x < 14

x < 7

Worked Example 4

Solve the double inequality: $-3 < 2x + 1 \leq 9$−3<2x+1≤9

Subtract 1 from all three parts: $-4 < 2x \leq 8$−4<2x≤8

Divide all three parts by 2: $-2 < x \leq 4$−2<x≤4

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Representing Inequalities on Number Lines

• An open circle (○) means the value is NOT included (< or >).
• A closed (filled) circle (●) means the value IS included $(\leq$(≤ or $\geq)$≥).
• Draw a line (or arrow) in the direction of the valid values.

Examples

• x > 3: open circle at 3, arrow pointing right $\to$→
• $x \leq 5$x≤5: closed circle at 5, arrow pointing left ←
• $-2 < x \leq 4$−2<x≤4: open circle at −2, closed circle at 4, line connecting them

Inequality on a Number Line — Symbol Map

flowchart LR
    INEQ[Inequality solved] --> SYM{Inequality symbol}
    SYM -->|x less than a| OPEN_R["Open circle at a<br/>arrow LEFT"]
    SYM -->|x greater than a| OPEN_L["Open circle at a<br/>arrow RIGHT"]
    SYM -->|x less than or equal a| CLOSED_R["Closed circle at a<br/>arrow LEFT"]
    SYM -->|x greater than or equal a| CLOSED_L["Closed circle at a<br/>arrow RIGHT"]
    SYM -->|a less than x less than b| BOTH_OPEN["Open at a, open at b<br/>line between"]
    SYM -->|a less than or equal x less than or equal b| BOTH_CLOSED["Closed at a, closed at b<br/>line between"]
    OPEN_R --> RULE["Open circle = NOT included<br/>Closed circle = included"]
    OPEN_L --> RULE
    CLOSED_R --> RULE
    CLOSED_L --> RULE
    BOTH_OPEN --> RULE
    BOTH_CLOSED --> RULE

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Listing Integer Values

Worked Example 5

List the integer values of n such that $-3 < n \leq 2$−3<n≤2.

n must be greater than −3 (not equal) and less than or equal to 2.

Integers: −2, −1, 0, 1, 2

Worked Example 6

List the integer values of x that satisfy both: x > −1 and 2x + 3 < 12.

From the second inequality: 2x < 9, so x < 4.5.

Combined: −1 < x < 4.5

Integer values: 0, 1, 2, 3, 4

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Graphical Inequalities [H]

To show a region on a graph that satisfies an inequality:

1. Draw the boundary line.
   • Solid line for $\leq$≤ or $\geq ($≥(the boundary is included).
   • Dashed line for < or > (the boundary is not included).
2. Test a point (usually the origin) to determine which side to shade.
3. Shade the required region (Edexcel convention: shade the region that satisfies the inequality, or label it clearly — always read the question).

Worked Example 7

Show the region satisfying: y < 2x + 1, $y \geq -1$y≥−1, x < 3.

1. Draw y = 2x + 1 as a dashed line (since <).
2. Draw y = −1 as a solid horizontal line (since $\geq)$≥).
3. Draw x = 3 as a dashed vertical line (since <).
4. Test (0, 0): Is 0 < 2(0) + 1 = 1? Yes ✓. Is $0 \geq -1$0≥−1? Yes ✓. Is 0 < 3? Yes ✓.

The origin is in the required region. Label the region R.

Worked Example 8

Write down the three inequalities that define the shaded region R, given the boundary lines y = x, y = 4, and x = 0 (the y-axis), where the shaded region is below y = x, below y = 4, and to the right of the y-axis.

Inequalities: $y \leq x$y≤x, $y \leq 4$y≤4, $x \geq 0$x≥0

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Quadratic Inequalities [H]

Method

1. Solve the corresponding quadratic equation (set the expression equal to 0).
2. Sketch the quadratic graph.
3. Use the sketch to determine which values of x satisfy the inequality.

Worked Example 9

Solve: $x^{2} - 5x + 6 < 0$x2−5x+6<0

Step 1: $x^{2} - 5x + 6 = 0 \to (x - 2)(x - 3) = 0 \to x = 2$x2−5x+6=0→(x−2)(x−3)=0→x=2 or x = 3

Step 2: The graph of $y = x^{2} - 5x + 6$y=x2−5x+6 is a U-shaped parabola crossing the x-axis at x = 2 and x = 3.

Step 3: The quadratic is below the x-axis (< 0) between the roots.

Answer: 2 < x < 3

Worked Example 10

Solve: $x^{2} + 2x - 8 \geq 0$x2+2x−8≥0

Step 1: $x^{2} + 2x - 8 = 0 \to (x + 4)(x - 2) = 0 \to x = -4$x2+2x−8=0→(x+4)(x−2)=0→x=−4 or x = 2

Step 2: U-shaped parabola crossing at −4 and 2.

Step 3: We want where the curve is above or on the $x-axis (\geq 0)$x−axis(≥0) — this is to the left of −4 and to the right of 2.

Answer: $x \leq -4$x≤−4 or $x \geq 2$x≥2

Worked Example 11

Solve: $2x^{2} - x - 6 > 0$2x2−x−6>0

$2x^{2} - x - 6 = 0 \to (2x + 3)(x - 2) = 0 \to x = -3/2$2x2−x−6=0→(2x+3)(x−2)=0→x=−3/2 or x = 2

U-shaped parabola, positive outside the roots.

Answer: x < −3/2 or x > 2

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Common Mistakes and Misconceptions

Mistake	Correct approach
Forgetting to flip the sign when dividing by a negative	Always reverse: if −2x > 6 then x < −3
Using = instead of the inequality symbol	Keep the inequality throughout your working
Open/closed circle confusion	< and $> \to open$>→open; $\leq$≤ and $\geq \to$≥→ closed
Writing $x^{2} - 5x + 6 < 0$x2−5x+6<0 as x < 2 or x < 3	It's a connected region: 2 < x < 3
Not sketching the quadratic for quadratic inequalities	Always sketch — it prevents sign errors

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Edexcel Exam Tips