Section B — Chapters 23–39

Algebra IV: Solving Linear Inequalities

Junior Cycle — 1st Year

✓By the end of this lesson students will be able to understand and use inequality symbols (<, >, ≤, ≥).
✓By the end of this lesson students will be able to represent inequalities on a number line.
✓By the end of this lesson students will be able to solve linear inequalities involving one variable.
✓By the end of this lesson students will be able to correctly handle the inequality sign when multiplying or dividing by a negative number.

Key concepts

Inequalities

An inequality is a mathematical statement that compares two expressions using an inequality symbol. Unlike an equation which states that two expressions are equal, an inequality states that one expression is greater than, less than, greater than or equal to, or less than or equal to another expression. The solution to an inequality is often a range of values, not just a single value.

Inequality Symbols

There are four main inequality symbols:\n\n* < means 'is less than'\n* > means 'is greater than'\n* ≤ means 'is less than or equal to'\n* ≥ means 'is greater than or equal to'

Graphing Inequalities on a Number Line

We can represent the solution set of an inequality on a number line. \n\n* For 'less than' (<) or 'greater than' (>) inequalities, we use an **open circle** at the boundary point to show that the point itself is *not* included in the solution.\n* For 'less than or equal to' (≤) or 'greater than or equal to' (≥) inequalities, we use a **closed circle** (or a filled-in dot) at the boundary point to show that the point *is* included in the solution.\n* An arrow is drawn from the circle in the direction of the solution set.

Solving Linear Inequalities

Solving linear inequalities is very similar to solving linear equations. You use inverse operations (addition, subtraction, multiplication, division) to isolate the variable. The key difference is when you multiply or divide both sides of an inequality by a negative number: you **must reverse the direction of the inequality sign**.

Key facts to remember

1The four inequality symbols are: < (less than), > (greater than), ≤ (less than or equal to), ≥ (greater than or equal to).
2Solving inequalities is similar to solving equations, using inverse operations.
3When multiplying or dividing both sides of an inequality by a negative number, you must reverse the direction of the inequality sign.
4On a number line, an open circle (o) is used for < or > (boundary point not included).
5On a number line, a closed circle (•) is used for ≤ or ≥ (boundary point included).
6The solution to an inequality is usually a range of values, not a single value.

Worked examples

Example 1

Solve the inequality x + 5 > 8 and represent the solution on a number line.

Ix + 5 > 8

IISubtract 5 from both sides to isolate x:

IIIx + 5 - 5 > 8 - 5

IVx > 3

VTo represent on a number line: Draw an open circle at 3 and an arrow pointing to the right (towards larger numbers).

Answer

x > 3

Since the inequality is 'greater than' (>), the number 3 itself is not included in the solution, so we use an open circle.

Example 2

Solve the inequality 3x - 2 ≤ 10 and represent the solution on a number line.

I3x - 2 ≤ 10

IIAdd 2 to both sides:

III3x - 2 + 2 ≤ 10 + 2

IV3x ≤ 12

VDivide both sides by 3 (a positive number):

VI3x / 3 ≤ 12 / 3

VIIx ≤ 4

VIIITo represent on a number line: Draw a closed circle at 4 and an arrow pointing to the left (towards smaller numbers).

Answer

x ≤ 4

Since the inequality is 'less than or equal to' (≤), the number 4 is included in the solution, so we use a closed circle.

Example 3

Solve the inequality -2x + 1 < 7 and represent the solution on a number line.

I-2x + 1 < 7

IISubtract 1 from both sides:

III-2x + 1 - 1 < 7 - 1

IV-2x < 6

VDivide both sides by -2 (a negative number). Remember to reverse the inequality sign:

VI-2x / -2 > 6 / -2

VIIx > -3

VIIITo represent on a number line: Draw an open circle at -3 and an arrow pointing to the right (towards larger numbers).

Answer

x > -3

This is a crucial step: when dividing by a negative number, the inequality sign MUST be reversed. If you forget this, your answer will be incorrect.

Common mistakes

✗Forgetting to reverse the inequality sign when multiplying or dividing by a negative number.
✗Incorrectly using an open circle instead of a closed circle (or vice versa) when graphing on a number line.
✗Confusing the direction of the arrow on the number line (e.g., drawing an arrow to the left for x > 5).
✗Treating an inequality like an equation and only finding a single value solution.

Exam tips

★Always double-check your work, especially when dealing with negative numbers and reversing the inequality sign.
★To verify your solution, pick a number from your solution set and substitute it back into the original inequality. It should make the inequality true.
★Draw your number lines clearly and accurately, paying attention to open/closed circles and arrow direction.
★Read the question carefully to see if it asks for the solution only, or also for a number line representation.

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