Inequalities On Graphs Quizzes

Introduction

Graphical inequalities are a vital topic in GCSE Maths, helping students visually represent solutions to inequalities and understand the relationship between variables. Mastery of inequalities on graphs allows students to solve linear and quadratic inequalities, interpret shaded regions, and apply these skills to real-world problems such as speed-distance-time, economics, and geometry.

Core Concepts

What are Inequalities?

An inequality compares two expressions and indicates that one is greater than, less than, greater than or equal to, or less than or equal to another. Common symbols include:

• \(>\) : greater than
• \(<\) : less than
• \(\geq\) : greater than or equal to
• \(\leq\) : less than or equal to

Example: \(2x + 3 < 7\) means that \(2x + 3\) is less than 7.

Linear Inequalities

A linear inequality involves a straight line and has the general form:

$$ ax + by \leq c \quad \text{or} \quad ax + by > c $$

Graphically, the solution is represented by shading the region of the plane that satisfies the inequality.

Quadratic Inequalities

Quadratic inequalities involve terms with \(x^2\) and can be written as:

$$ ax^2 + bx + c \geq 0 \quad \text{or} \quad ax^2 + bx + c < 0 $$

The solutions are represented by regions above or below the parabola, depending on the inequality sign.

Boundary Lines

When graphing inequalities:

• Use a solid line for \(\geq\) or \(\leq\) because the points on the line satisfy the inequality.
• Use a dashed line for \(>\) or \(<\) because the points on the line do not satisfy the inequality.

Rules & Steps

1. Graphing a Linear Inequality

1. Rewrite in the form \(y \leq mx + c\) or \(y \geq mx + c\).
2. Draw the boundary line \(y = mx + c\) (solid or dashed depending on the inequality).
3. Choose a test point not on the line (commonly \((0,0)\)) to determine which side of the line satisfies the inequality.
4. Shade the appropriate region of the graph.

Example:

$$ y < 2x + 3 $$

• Boundary: \(y = 2x + 3\) (dashed line because <)
• Test point \((0,0)\): \(0 < 2(0) + 3 \Rightarrow 0 < 3\) ✓ True, shade region containing (0,0)

2. Solving and Graphing Systems of Inequalities

1. Graph each inequality separately.
2. The solution set is the intersection of the shaded regions.
3. Ensure clear labeling and consistent shading patterns.

Example:

$$ \begin{cases} y \geq x - 1 \\ y \leq -x + 3 \end{cases} $$

The solution is the overlapping shaded region between the two lines.

3. Quadratic Inequalities

1. Rewrite in standard form: \(ax^2 + bx + c \geq 0\) or \(ax^2 + bx + c < 0\).
2. Find roots of the corresponding quadratic equation \(ax^2 + bx + c = 0\).
3. Plot the parabola \(y = ax^2 + bx + c\).
4. Determine regions where the parabola lies above or below the x-axis depending on the inequality sign.
5. Shade the correct region; for \(\geq\) or \(\leq\), include roots with solid dots; for \(>\) or \(<\), use open dots.

Example:

$$ x^2 - 4x + 3 > 0 $$

• Find roots: \(x^2 - 4x + 3 = 0 \Rightarrow (x-1)(x-3)=0 \Rightarrow x = 1, 3\)
• Graph parabola \(y = x^2 - 4x + 3\) opening upwards.
• Solution: \(x < 1\) or \(x > 3\) (regions above x-axis)

Worked Examples

1. Linear inequality: \(y \geq -x + 2\)
   • Boundary: \(y = -x + 2\) (solid line)
   • Test point \((0,0)\): \(0 \geq -0 + 2 \Rightarrow 0 \geq 2\) ✗ False, shade opposite side
2. System of inequalities: $$ \begin{cases} y \leq 2x + 1 \\ y \geq x - 1 \end{cases} $$
   • Draw \(y = 2x + 1\) (dashed line) and \(y = x - 1\) (solid line)
   • Shade below first line, above second line
   • Intersection is the solution region
3. Quadratic inequality: \(x^2 - 5x + 6 \leq 0\)
   • Factor: \((x-2)(x-3) \leq 0\)
   • Roots: \(x = 2, 3\)
   • Parabola opens upwards (\(a>0\)), inequality \(\leq 0\) → shade between roots
   • Solution: \(2 \leq x \leq 3\)
4. Real-world application: A company has constraints on production: $$ \begin{cases} x + y \leq 10 \\ x \geq 2 \\ y \geq 1 \end{cases} $$
   • Graph each inequality
   • Solution: feasible region bounded by lines and axes
5. Higher-tier example: \(y > x^2 - 4x + 3\)
   • Roots: \(x^2 - 4x + 3 = 0 \Rightarrow x=1,3\)
   • Parabola opens upwards, shade region above parabola

Common Mistakes

• Using a solid line for \(>\) or \(<\) instead of dashed line.
• Incorrect shading; always test a point to confirm the correct side.
• Neglecting intersections when solving systems of inequalities.
• Confusing linear and quadratic inequalities and shading incorrectly.
• Forgetting to include boundary points when using \(\geq\) or \(\leq\).

Applications

• Maximizing or minimizing constraints in business or resource allocation problems.
• Physics: representing speed, distance, or time limitations.
• Geometry: defining feasible regions for shapes or areas.
• Economics: visualizing budget or production constraints.

Strategies & Tips

• Always rewrite inequalities in slope-intercept form \(y = mx + c\) for easier graphing.
• Use a test point to determine the correct region to shade.
• For systems, carefully identify the overlapping shaded region.
• Check all roots or intersection points when dealing with quadratics.
• Practice drawing accurate axes and scaling to avoid misinterpretation.

Summary

Graphical inequalities provide a visual method to understand and solve inequalities in GCSE Maths. By combining skills in algebra, graphing, and logical reasoning, students can tackle linear, quadratic, and system inequalities effectively. Regular practice with shading regions, testing points, and interpreting intersections will enhance both confidence and exam performance. Attempt the quizzes and exercises to strengthen your understanding and mastery of inequalities on graphs.