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Algebra - Linear Inequalities

Grade 8IGCSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Inequality Symbols: Use << (less than), >> (greater than), ≤\leq (less than or equal to), and ≥\geq (greater than or equal to).

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Solving Inequalities: Follow the same steps as solving linear equations (using inverse operations) to isolate the variable.

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The Negative Rule: When multiplying or dividing both sides of an inequality by a negative number, the direction of the inequality sign must be reversed.

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Number Line Representation: Use an open circle (○) for << and >> (exclusive) and a solid circle (●) for ≤\leq and ≥\geq (inclusive).

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Solution Sets: The solution to an inequality is a range of values, rather than a single number.

📐Formulae

If a>ba > b, then a+c>b+ca + c > b + c

If a>ba > b and c>0c > 0, then ac>bcac > bc

If a>ba > b and c<0c < 0, then ac<bcac < bc (Sign Reversal)

x∈R represents all real numbers in the solutionx \in \mathbb{R} \text{ represents all real numbers in the solution}

💡Examples

Problem 1:

Solve the inequality: 3x+4<193x + 4 < 19

Solution:

3x<153x < 15 \ x<5x < 5

Explanation:

Subtract 4 from both sides to get 3x<153x < 15. Then, divide both sides by 3. Since 3 is positive, the inequality sign stays the same.

Problem 2:

Solve the inequality: 10−2x≤1610 - 2x \leq 16

Solution:

−2x≤6-2x \leq 6 \ x≥−3x \geq -3

Explanation:

First, subtract 10 from both sides to get −2x≤6-2x \leq 6. Next, divide by -2. Because we are dividing by a negative number, the ≤\leq sign flips to ≥\geq.

Problem 3:

Solve 4(x−2)>2x+64(x - 2) > 2x + 6 and represent on a number line.

Solution:

4x−8>2x+64x - 8 > 2x + 6 \ 2x−8>62x - 8 > 6 \ 2x>142x > 14 \ x>7x > 7

Explanation:

Expand the brackets first. Then, collect like terms by subtracting 2x2x from both sides and adding 8 to both sides. Finally, divide by 2. On a number line, this is represented by an open circle at 7 with an arrow pointing to the right.