Linear Inequalities Revision Notes for Leaving Cert Mathematics

Linear Inequalities

What are linear inequalities?

A linear inequality is a mathematical statement that compares two algebraic expressions using inequality symbols rather than an equals sign. Unlike equations where both sides are equal, inequalities show that one side is greater than, less than, greater than or equal to, or less than or equal to the other side.

Understanding the fundamental difference between equations and inequalities is essential for mastering this topic. While equations represent exact relationships, inequalities describe ranges of possible values.

For example:

$2x + 4 = 6$ is an equation because both sides are equal
$2x + 4 > 6$ is an inequality because both sides are not equal

Inequality symbols

There are four main inequality symbols you need to know:

Symbol	Meaning
$>$	is greater than
$\geq$	is greater than or equal to
$<$	is less than
$\leq$	is less than or equal to

These symbols help us express relationships between numbers and algebraic expressions where one value is larger or smaller than another. Mastering these symbols is essential for understanding how inequalities work and how to interpret their solutions correctly.

Rules for solving inequalities

The process of solving inequalities follows very similar steps to solving equations. The algebraic manipulation techniques you've learned for equations apply directly to inequalities, making them relatively straightforward to solve once you understand the key principles.

chatImportant

The inequality sign is reversed when both sides are multiplied or divided by the same negative number.

This is the most critical rule to remember when solving inequalities. Many students make errors by forgetting this essential step.

For example:

We know that $3 < 5$
But when we multiply both sides by $-1$ : $3 \times (-1) > 5 \times (-1)$
This gives us $-3 > -5$

Notice how the $<$ symbol changed to $>$ when we multiplied by the negative number.

Number sets and the number line

Before solving inequalities, it's helpful to understand the different types of numbers and how they appear on the number line. The type of number set specified in a problem will determine which values can be included in your solution.

Natural numbers (N)

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The set of natural numbers is $N = \{1, 2, 3, 4, ...\}$

These are the counting numbers starting from 1 and continuing infinitely. On the number line, they appear as individual points starting from 1. Remember that natural numbers do not include zero or negative numbers.

Integers (Z)

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The set of integers is $Z = \{..., -3, -2, -1, 0, 1, 2, 3, ...\}$

Integers include all whole numbers, both positive and negative, as well as zero. They extend infinitely in both directions on the number line and represent discrete points rather than continuous values.

Real numbers (R)

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The set of real numbers contains all numbers on the number line. This includes natural numbers, integers, fractions, and decimal numbers. Real numbers are represented on the number line by a bold line indicating that all numbers are included.

Representing solutions on the number line

When showing inequality solutions on a number line, proper notation is crucial for communicating your results clearly:

A full circle (●) indicates the number is included in the solution
An empty circle (○) indicates the number is not included in the solution

For example, if $x \geq 4$ , we draw a full circle at 4 and shade to the right. If $x < -2$ , we draw an empty circle at -2 and shade to the left.

Solving linear inequalities step-by-step

The systematic approach to solving linear inequalities involves the same algebraic techniques used for equations, with careful attention to the sign reversal rule for negative operations.

lightbulbExample

Worked Example: Basic Inequality

Solve $5 - 2x < 9$ where $x \in Z$ and illustrate the solution on the number line.

Step 1: Subtract 5 from both sides $5 - 2x - 5 < 9 - 5$ $-2x < 4$

Step 2: Divide both sides by -2 (remember to reverse the inequality sign!) $\frac{-2x}{-2} > \frac{4}{-2}$ $x > -2$

Step 3: Illustrate on the number line The solution shows all integers greater than -2, so $x \in \{-1, 0, 1, 2, 3, ...\}$

lightbulbExample

Worked Example: Finding Intersection of Solution Sets

Find the solution set A of $3x + 1 \leq 2x + 5$ where $x \in R$ Find the solution set B of $\frac{1}{3} - 2x \leq \frac{25}{3}$ where $x \in R$
Find $A \cap B$ and illustrate on the number line.

Solving inequality A: $3x + 1 \leq 2x + 5$ $3x - 2x + 1 \leq 2x - 2x + 5$ $x + 1 \leq 5$ $x + 1 - 1 \leq 5 - 1$
$x \leq 4$

Solving inequality B: $\frac{1}{3} - 2x \leq \frac{25}{3}$ $1 - 6x \leq 25$ (multiplying by 3) $1 - 1 - 6x \leq 25 - 1$ $-6x \leq 24$ $\frac{-6x}{-6} \geq \frac{24}{-6}$ (dividing by -6, so reverse the sign) $x \geq -4$

Combining the results: We need $x \leq 4$ AND $x \geq -4$ Therefore: $A \cap B = \{x: -4 \leq x \leq 4\}$

On the number line, this appears as a line segment from -4 to 4, with both endpoints included (shown with full circles).

Key solving strategies

Developing a systematic approach to solving linear inequalities will help you avoid common mistakes and work more efficiently. These strategies build on your existing equation-solving skills while accounting for the unique aspects of inequalities.

Isolate the variable by performing the same operations on both sides
Watch for negative coefficients - always reverse the inequality sign when multiplying or dividing by a negative number
Check your answer by substituting a value from your solution set back into the original inequality
Use appropriate notation for the number set specified in the question (N, Z, or R)

Common exam tips

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Essential Exam Strategies:

Always state your final answer clearly using proper mathematical notation
When graphing on a number line, use the correct circle notation (full for ≤ or ≥, empty for < or >)
Remember that natural numbers start from 1, not 0
Double-check your work when dealing with negative coefficients
Practice converting between different forms of the same inequality

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Key Points to Remember:

Linear inequalities compare algebraic expressions using symbols like >, ≥, <, ≤ instead of equals signs
The inequality sign reverses when multiplying or dividing both sides by a negative number
Number line representations use full circles for "or equal to" and empty circles for strict inequalities
Different number sets (N, Z, R) affect which values can be included in your solution
Always check your solutions by substituting values back into the original inequality

Linear Inequalities (Leaving Cert Mathematics): Revision Notes