Linear inequalities: solving and representing on a number line or graph

What you'll learn

Linear inequalities form a crucial component of the CIE IGCSE Mathematics syllabus, appearing regularly in Paper 2 and Extended Paper 4 examinations. This topic extends your knowledge of linear equations by introducing inequality symbols and requiring you to represent solution sets both algebraically and graphically. Understanding how to manipulate inequalities while preserving their truth, and accurately depicting solutions on number lines or coordinate graphs, are essential skills tested across multiple question types.

Key terms and definitions

Inequality — a mathematical statement showing that two expressions are not equal, using symbols <, >, ≤, or ≥ instead of an equals sign.

Linear inequality — an inequality containing variables to the first power only, such as 3x + 5 > 11 or 2y - 7 ≤ 15.

Solution set — the complete collection of all values that satisfy an inequality, often represented as an interval or region.

Boundary line — the line representing the equation formed when an inequality sign is replaced with an equals sign; determines the edge of a solution region on a graph.

Open circle (○) — notation used on a number line to show that a boundary value is not included in the solution set (for < or > inequalities).

Closed circle (●) — notation used on a number line to show that a boundary value is included in the solution set (for ≤ or ≥ inequalities).

Test point — a coordinate used to determine which side of a boundary line satisfies an inequality in two variables.

Feasible region — the area on a graph where all inequalities in a system are simultaneously satisfied, commonly tested in linear programming contexts.

Core concepts

Understanding inequality notation

The four main inequality symbols convey different relationships:

• < (less than): values strictly smaller than the boundary
• > (greater than): values strictly larger than the boundary
• ≤ (less than or equal to): values up to and including the boundary
• ≥ (greater than or equal to): values from the boundary upwards

When reading inequalities, the symbol always "points" to the smaller value. The statement x < 5 is equivalent to 5 > x; both mean x is less than 5.

Solving linear inequalities in one variable

The process mirrors solving linear equations with one critical exception: when multiplying or dividing both sides by a negative number, reverse the inequality sign.

Standard solving steps:

1. Expand any brackets using the distributive property
2. Collect all terms containing the variable on one side
3. Collect all constant terms on the other side
4. Isolate the variable by performing inverse operations
5. If dividing or multiplying by a negative, flip the inequality symbol
6. Write the solution in the form x < a, x > a, x ≤ a, or x ≥ a

Example process:

• Solve 5 - 2x ≥ 11
• Subtract 5: -2x ≥ 6
• Divide by -2 (reverse sign): x ≤ -3

The reversal rule stems from the number line: if x < y, then multiplying both by -1 gives -x > -y (the order flips).

Representing solutions on a number line

Number line representation provides a visual summary of the solution set:

For simple inequalities:

• Draw a horizontal line with appropriate scale markings
• Locate the boundary value
• Use ○ (open circle) for strict inequalities (< or >)
• Use ● (closed circle) for inclusive inequalities (≤ or ≥)
• Shade or draw an arrow in the direction of values that satisfy the inequality

For compound inequalities (e.g., -2 < x ≤ 5):

• Mark both boundary points
• Use appropriate circle notation for each end
• Shade the region between the boundaries
• This represents all values satisfying both conditions simultaneously

Representing linear inequalities on coordinate graphs

For inequalities in two variables (x and y), solutions form regions on the coordinate plane:

Step-by-step graphing procedure:

1. Convert to boundary equation: Replace the inequality symbol with =
2. Plot the boundary line: Use standard methods (table of values, intercepts, or gradient-intercept form)
3. Determine line style:
   • Solid line for ≤ or ≥ (boundary included in solution)
   • Dashed line for < or > (boundary excluded from solution)
4. Identify the solution region: Use a test point (often the origin (0,0) if not on the line)
5. Shade appropriately: Shade the side containing points that satisfy the inequality

Test point method: Substitute the test point coordinates into the original inequality. If the statement is true, shade the region containing that point. If false, shade the opposite side.

For inequalities like y > 2x + 3, a quick method exists: the > symbol means shade above the line; < means shade below (when y is isolated on the left).

Solving systems of linear inequalities

CIE IGCSE examinations frequently test systems of multiple inequalities simultaneously, particularly in linear programming contexts.

Procedure for systems:

1. Graph each inequality individually on the same coordinate plane
2. Use different line styles or label each boundary clearly
3. Identify the feasible region where all shaded areas overlap
4. This overlapping region satisfies all inequalities simultaneously
5. Clearly indicate this region (often by labelling it with a letter like R)

Additional constraints commonly tested:

• x ≥ 0 and y ≥ 0 (restricts solutions to the first quadrant)
• Integer solutions only (requires identifying lattice points within the region)
• Optimization problems (finding maximum/minimum values at vertices)

Inequality manipulation rules

Understanding which operations preserve inequality direction and which reverse it:

Operations that preserve direction:

• Adding or subtracting the same value from both sides
• Multiplying or dividing both sides by a positive number
• Squaring both sides (only when both sides are positive)

Operations that reverse direction:

• Multiplying or dividing both sides by a negative number
• Taking reciprocals of both sides (when both sides have the same sign)

Example of reciprocal reversal: If 2 < 5, then 1/2 > 1/5.

Worked examples

Example 1: Solving and representing a linear inequality

Question: Solve the inequality 3(2x - 1) ≤ 4x + 9 and represent your solution on a number line.

Solution:

Expand the brackets: 6x - 3 ≤ 4x + 9

Subtract 4x from both sides: 2x - 3 ≤ 9

Add 3 to both sides: 2x ≤ 12

Divide by 2: x ≤ 6

Number line representation: Draw a number line with values around 6. Place a closed circle (●) at 6 and shade/arrow pointing left to indicate all values less than or equal to 6.

[Mark scheme: 1 mark for correct manipulation, 1 mark for x ≤ 6, 1 mark for correct number line with closed circle and correct shading]

Example 2: Graphical inequality with two variables

Question: On a coordinate grid, show by shading the region that satisfies all three inequalities:

• y ≥ 1
• x + y ≤ 5
• y < 2x

Label the region R.

Solution:

For y ≥ 1:

• Boundary line: y = 1 (horizontal line through y = 1)
• Solid line (≥ includes boundary)
• Shade above the line

For x + y ≤ 5:

• Boundary equation: x + y = 5, or y = 5 - x
• When x = 0, y = 5; when y = 0, x = 5
• Solid line connecting (0, 5) and (5, 0)
• Test point (0, 0): 0 + 0 ≤ 5 ✓ (true)
• Shade below/left of the line

For y < 2x:

• Boundary equation: y = 2x (passes through origin with gradient 2)
• Dashed line (< excludes boundary)
• Test point (1, 0): 0 < 2(1) = 2 ✓ (true)
• Shade below the line

Final region R: The area where all three shaded regions overlap, bounded by these three lines.

[Mark scheme: 1 mark for each correctly drawn boundary line, 1 mark for correct line style (solid/dashed), 2 marks for correctly identified region R]

Example 3: Solving inequality requiring sign reversal

Question: Solve the inequality 7 - 3x > 16, giving your answer in the form x ⋈ k where ⋈ is an inequality symbol and k is an integer.

Solution:

Start with: 7 - 3x > 16

Subtract 7 from both sides: -3x > 9

Divide both sides by -3 (reverse inequality sign): x < -3

Check: Test x = -4 (which is less than -3): 7 - 3(-4) = 7 + 12 = 19, and 19 > 16 ✓

[Mark scheme: 1 mark for correct rearrangement to -3x > 9, 1 mark for x < -3 with correctly reversed inequality]

Common mistakes and how to avoid them

• Forgetting to reverse the inequality when dividing/multiplying by negatives — Always remember: if you divide or multiply by a negative number, flip < to > or ≤ to ≥. Write a reminder at the top of your working. Check your answer makes sense by substituting a test value.

• Using the wrong circle notation on number lines — Open circles (○) are for strict inequalities (< or >), closed circles (●) are for inclusive inequalities (≤ or ≥). The word "equal" in "less than or equal to" signals a closed circle.

• Shading the wrong side of a boundary line — Always use a test point (usually (0,0) if convenient) and substitute into the original inequality to determine which side to shade. Don't rely on guessing.

• Drawing solid lines when dashed is required (or vice versa) — Solid lines are for ≤ and ≥ (boundary included); dashed lines are for < and > (boundary excluded). In exam questions, this distinction often carries a specific mark.

• Misinterpreting compound inequalities like -3 < x ≤ 2 — This means x is greater than -3 AND less than or equal to 2 simultaneously. On a number line, use an open circle at -3, a closed circle at 2, and shade the region between them. It does not mean two separate regions.

• Incorrectly handling inequalities with y on the right side — If given x + 3 < y, rearrange to y > x + 3 before graphing to clearly see "greater than means shade above." Alternatively, test points carefully to determine the correct region without rearranging.

Exam technique for Linear inequalities: solving and representing on a number line or graph

• Command words and what they require: "Solve" requires algebraic manipulation to isolate the variable; "represent" or "show" requires a number line diagram; "shade the region" or "indicate the region" requires a graph with correct boundary lines and clear shading. The word "show" often requires visible working, not just the final answer.

• Accuracy in graphical work: Use a ruler for all straight lines. Plot at least two points to establish each boundary line accurately. When asked to shade a region, use light, consistent shading or clear diagonal lines—examiners must be able to see your boundaries clearly. Always label required regions with the specified letter.

• Showing reversal explicitly: When your working requires reversing an inequality sign, examiners look for clear evidence. Write the step where you divide/multiply by a negative, then explicitly show the reversed symbol in the next line. This often secures a method mark even if arithmetic errors exist elsewhere.

• Mark allocation patterns: One inequality typically awards 2-3 marks (1 for method, 1-2 for correct answer and representation). Systems of inequalities often award 4-6 marks (1 per boundary line, 1 for line style, 1-2 for correct region identification). Expect linear programming questions to reach 6-8 marks when optimization is included.

Quick revision summary

Linear inequalities extend equation-solving by using <, >, ≤, or ≥ symbols. Solve by isolating the variable, remembering to reverse the inequality when multiplying or dividing by negatives. Represent solutions on number lines using open circles (○) for strict inequalities and closed circles (●) for inclusive ones. Graph two-variable inequalities by plotting the boundary line (solid for ≤/≥, dashed for </>) and shading the correct region using test points. Systems of inequalities create feasible regions where multiple conditions overlap. Master the reversal rule and accurate graphing for examination success.