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Algebra and Graphs - Linear inequalities

Grade 11IGCSE

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

🔑Concepts

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Definition of inequality symbols: < (less than), > (greater than), ≤ (less than or equal to), ≥ (greater than or equal to).

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Solving linear inequalities: Follow the same steps as linear equations, but reverse the inequality sign when multiplying or dividing by a negative number.

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Number line representation: Use an open circle for < or > and a closed (solid) circle for ≤ or ≥.

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Graphical representation on a Cartesian plane: Use a dashed line for strict inequalities (<, >) and a solid line for inclusive inequalities (≤, ≥).

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Region shading: Identify the 'wanted' region by testing a point (usually (0,0)) not on the boundary line.

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System of inequalities: The solution is the overlapping region (feasible region) that satisfies all given inequalities simultaneously.

📐Formulae

ax+b<cax + b < c

If ax>b and a<0, then x<ba (Sign Reversal Rule)If \ ax > b \ and \ a < 0, \ then \ x < \frac{b}{a} \text{ (Sign Reversal Rule)}

y>mx+c (Region above the line)y > mx + c \text{ (Region above the line)}

y<mx+c (Region below the line)y < mx + c \text{ (Region below the line)}

x=k (Vertical boundary line)x = k \text{ (Vertical boundary line)}

y=k (Horizontal boundary line)y = k \text{ (Horizontal boundary line)}

💡Examples

Problem 1:

Solve the inequality: 5−2x≤115 - 2x \leq 11.

Solution:

−2x≤6  ⟹  x≥−3-2x \leq 6 \implies x \geq -3

Explanation:

Subtract 5 from both sides to get -2x ≤ 6. When dividing by -2, the inequality sign must be flipped from ≤ to ≥.

Problem 2:

Represent the region defined by y≥2x−4y \geq 2x - 4 on a graph.

Solution:

Draw the line y=2x−4y = 2x - 4 as a solid line. Shade the area above the line.

Explanation:

The line is solid because of the 'equal to' part of the symbol (≥). Testing (0,0): 0≥2(0)−40 \geq 2(0) - 4 results in 0≥−40 \geq -4, which is true, so the side containing the origin is shaded.

Problem 3:

Find the integer values of xx that satisfy: −3<2x+1≤7-3 < 2x + 1 \leq 7.

Solution:

−4<2x≤6  ⟹  −2<x≤3-4 < 2x \leq 6 \implies -2 < x \leq 3. Integers: {−1,0,1,2,3}\{-1, 0, 1, 2, 3\}.

Explanation:

Subtract 1 from all parts of the inequality, then divide all parts by 2. The solution includes integers greater than -2 and up to (and including) 3.