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Algebra - Linear Inequalities

Grade 9IGCSE

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

🔑Concepts

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Inequality Symbols: Understanding < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to).

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Solving Linear Inequalities: The process is similar to 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 < or > and a solid circle (●) for ≤ or ≥.

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Integer Solutions: Identifying the set of whole numbers or integers that satisfy a given inequality range.

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Graphical Inequalities: Representing inequalities on a Cartesian plane where the line is dashed for strict inequalities (<, >) and solid for inclusive inequalities (≤, ≥).

📐Formulae

If ax+b>cax + b > c, then ax>c−bax > c - b

If x/k<ax/k < a and k>0k > 0, then x<akx < ak

If −ax<b-ax < b, then x>−b/ax > -b/a (Sign reversal rule)

Compound Inequality: a<x<ba < x < b represents all values of xx between aa and bb excluding the endpoints.

💡Examples

Problem 1:

Solve the inequality: 3x−7≤113x - 7 \leq 11.

Solution:

3x≤183x \leq 18 x≤6x \leq 6

Explanation:

Add 7 to both sides of the inequality to isolate the term with x. Then, divide both sides by 3. Since 3 is positive, the inequality sign remains the same.

Problem 2:

Solve the inequality: 10−2x<410 - 2x < 4.

Solution:

−2x<−6-2x < -6 x>3x > 3

Explanation:

First, subtract 10 from both sides. This leaves −2x<−6-2x < -6. When dividing both sides by -2, the inequality sign must be flipped from 'less than' to 'greater than'.

Problem 3:

List the integer values of nn such that −2≤n<3-2 \leq n < 3.

Solution:

{−2,−1,0,1,2}\{-2, -1, 0, 1, 2\}

Explanation:

The symbol ≤\leq means -2 is included in the set. The symbol << means 3 is excluded. We list all integers starting from -2 up to, but not including, 3.

Problem 4:

Solve the double inequality: 5<2x+1≤135 < 2x + 1 \leq 13.

Solution:

4<2x≤124 < 2x \leq 12 2<x≤62 < x \leq 6

Explanation:

Subtract 1 from all three parts of the inequality to get 4<2x≤124 < 2x \leq 12. Then, divide all parts by 2 to isolate xx, resulting in the final range.