Solving Linear Inequalities Revision Notes for Grade 10 NSC Matric Mathematics

Solving Linear Inequalities

What are linear inequalities?

A linear inequality is similar to a linear equation, but instead of an equals sign, it uses inequality symbols. The key characteristic is that the highest power of the variable is 1, just like in linear equations.

Examples of linear inequalities:

$2x + 2 \leq 1$
$\frac{2-x}{3x+1} \geq 2$
$\frac{4}{3}x - 6 < 7x + 2$

The main difference between equations and inequalities is that equations have one specific solution, while inequalities have a solution set - a range of values that satisfy the inequality.

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Unlike linear equations which give you a single point as an answer, linear inequalities provide a range of values. This is why we represent solutions as intervals or regions on number lines rather than individual points.

Solving methods

The methods for solving linear inequalities are very similar to those used for linear equations. You can add, subtract, multiply, and divide both sides of an inequality, just like with equations. However, there is one crucial difference you must remember.

The negative rule

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Critical Rule: When you multiply or divide both sides of an inequality by a negative number, you must reverse the direction of the inequality sign.

For example, if you have $-x > 3$ and you multiply both sides by $-1$ , the inequality becomes $x < -3$ (notice how the > becomes <).

Let's see why this rule exists. We know that $8 > 6$ . If we multiply both sides by $-2$ , we get $-16$ and $-12$ . Since $-16 < -12$ , the inequality sign must flip to maintain the true relationship.

Representing solutions on number lines

Unlike equations that have a single point as a solution, inequalities have solution sets that we represent on number lines using different symbols:

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Number Line Symbols:

Filled/closed dot: Used when the value is included in the solution (≤ or ≥)
Open/hollow circle: Used when the value is not included in the solution (< or >)
Arrow or shading: Shows the direction of all values that satisfy the inequality

Interval notation

Interval notation is a mathematical way to write solution sets using brackets and parentheses.

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Key rules for interval notation:

Round brackets ( ): The endpoint is NOT included
Square brackets [ ]: The endpoint IS included
Infinity symbols: Always use round brackets with ∞ or -∞

Examples:

$(4; 12)$ means all numbers greater than 4 but less than 12
$(-∞; -1)$ means all numbers less than -1
$[1; 13)$ means all numbers from 1 (including 1) up to but not including 13

Worked examples

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Example 1: Basic linear inequality

Question: Solve $6 - r > 2$

Solution: Step 1: Rearrange to isolate the variable term

$6 - r > 2$
$-r > 2 - 6$
$-r > -4$

Step 2: Multiply both sides by -1 and reverse the inequality sign

$r < 4$

Step 3: Represent on a number line

The solution shows all values less than 4, with an open circle at 4.

Step 4: Write in interval notation

$(-∞; 4)$

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Example 2: Linear inequality with brackets

Question: Solve $4q + 3 < 2(q + 3)$

Solution: Step 1: Expand the brackets

$4q + 3 < 2(q + 3)$
$4q + 3 < 2q + 6$

Step 2: Rearrange to collect like terms

$4q + 3 < 2q + 6$
$4q - 2q < 6 - 3$
$2q < 3$

Step 3: Divide both sides by 2

$q < \frac{3}{2}$

Step 4: Write in interval notation

$(-∞; \frac{3}{2})$

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Example 3: Compound inequality

Question: Solve $5 \leq x + 3 < 8$

Solution: Step 1: Subtract 3 from all parts of the inequality

$5 - 3 \leq x + 3 - 3 < 8 - 3$
$2 \leq x < 5$

Step 2: The solution shows x can be any value from 2 (including 2) up to but not including 5.

Step 3: Write in interval notation

$[2; 5)$

Common exam tips

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Watch out for these common mistakes:

Forgetting to flip the inequality sign when multiplying or dividing by negative numbers
Mixing up round and square brackets in interval notation
Not representing the solution correctly on a number line

Exam strategy:

Always check your final answer by substituting a test value
Remember that inequality solutions are ranges, not single points
When graphing, use the correct symbols (open circles vs closed dots)

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Key Points to Remember:

Linear inequalities have solution sets (ranges) rather than single solutions
Always reverse the inequality sign when multiplying or dividing by negative numbers
Round brackets ( ) exclude endpoints, square brackets [ ] include endpoints
Use open circles for < or > and filled dots for ≤ or ≥ on number lines
Check your work by testing values in the original inequality

Solving Linear Inequalities (Grade 10 NSC Matric Mathematics): Revision Notes