# IGCSE Mathematics Higher: Inequalities

## Scheme of work: IGCSE Foundation: Year 11: Term 1: Inequalities

#### Prerequisite Knowledge

1. Solving Linear Inequalities
• Concept: Understanding how to solve linear inequalities and represent their solutions on a number line.
• Example: $\text{Solve the inequality } 2x – 3 < 7.$ $2x - 3 < 7 \rightarrow 2x < 10 \rightarrow x < 5.$ $\text{Solution: } x < 5 \text{, represented on a number line as an open circle at } 5 \text{ with a line extending to the left.}$
2. Plotting Straight Line Graphs
• Concept: Knowing how to plot straight line graphs using the slope-intercept form $$y = mx + c$$ or by finding and plotting points.
• Example: $\text{Plot the graph of } y = 2x + 3.$ $\text{Find points by substituting } x \text{ values: }$ $x = -1 \rightarrow y = 2(-1) + 3 = 1$ $x = 0 \rightarrow y = 3$ $x = 1 \rightarrow y = 5$ $\text{Plot the points and draw the line.}$
3. Solving Quadratic Equations by Factorisation
• Concept: Understanding how to solve quadratic equations by factorising the quadratic expression.
• Example: $\text{Solve } x^2 – 5x + 6 = 0 \text{ by factorisation.}$ $x^2 – 5x + 6 = (x – 2)(x – 3) = 0$ $\text{So, } x – 2 = 0 \text{ or } x – 3 = 0 \rightarrow x = 2 \text{ or } x = 3.$
• Concept: Knowing how to sketch the graph of a quadratic equation by identifying its key features such as the vertex, axis of symmetry, and x-intercepts.
• Example: $\text{Sketch the graph of } y = x^2 – 5x + 6.$ $\text{Factorise: } y = (x – 2)(x – 3)$ $\text{X-intercepts at } x = 2 \text{ and } x = 3.$ $\text{Vertex: } x = \frac{-b}{2a} = \frac{5}{2} = 2.5 \rightarrow y = (2.5)^2 – 5(2.5) + 6 = -0.25$ $\text{Plot these points and sketch the parabola.}$

#### Success Criteria

1. Solve Quadratic Inequalities in One Unknown
• Objective: Solve quadratic inequalities by finding the critical points and testing the intervals to determine the solution set.
• Example: $\text{Solve the inequality } x^2 – 5x + 6 < 0.$ $\text{Factorise: } (x - 2)(x - 3) < 0.$ $\text{Critical points: } x = 2 \text{ and } x = 3.$ $\text{Test intervals: } \begin{cases} x < 2 \rightarrow (1 - 2)(1 - 3) = 1 \cdot (-2) = -2 & \text{(negative)} \\ 2 < x < 3 \rightarrow (2.5 - 2)(2.5 - 3) = 0.5 \cdot (-0.5) = -0.25 & \text{(negative)} \\ x > 3 \rightarrow (4 – 2)(4 – 3) = 2 \cdot 1 = 2 & \text{(positive)} \end{cases}$ $\text{Solution: } 2 < x < 3.$
2. Represent the Solution Set of a Quadratic Inequality on a Number Line
• Objective: Accurately represent the solution set of a quadratic inequality on a number line.
• Example: $\text{For } x^2 – 5x + 6 < 0, \text{ the solution set } 2 < x < 3 \text{ is represented on the number line as an open interval between 2 and 3.}$
3. Identify Harder Examples of Regions Defined by Linear Inequalities
• Objective: Identify and represent regions defined by multiple linear inequalities, including more complex cases.
• Example: $\text{Find the region defined by } y \geq 2x + 1 \text{ and } y \leq -x + 4.$ $\text{Plot both lines and shade the region where } y \geq 2x + 1 \text{ and } y \leq -x + 4 \text{ intersect.}$
4. Interpret Solutions Graphically
• Objective: Accurately interpret the solutions of quadratic and linear inequalities by examining their graphical representations.
• Example: $\text{For } x^2 – 5x + 6 < 0 \text{, plot the parabola and identify the interval where the graph is below the x-axis.}$ $\text{For } y \geq 2x + 1 \text{ and } y \leq -x + 4 \text{, identify the overlapping shaded region.}$

#### Key Concepts

1. Solve Quadratic Inequalities in One Unknown
• Concept: Quadratic inequalities can be solved by finding the roots of the corresponding quadratic equation and testing intervals to determine where the inequality holds true.
• Example: $\text{For } x^2 – 5x + 6 < 0, \text{ factorise to } (x - 2)(x - 3) < 0.$ $\text{Find the roots } x = 2 \text{ and } x = 3 \text{, then test the intervals around the roots.}$
2. Represent the Solution Set of a Quadratic Inequality on a Number Line
• Concept: The solution set of a quadratic inequality can be represented on a number line by using open or closed circles to indicate whether the endpoints are included or excluded, and shading the interval where the inequality holds.
• Example: $\text{For } 2 < x < 3 \text{, use open circles at 2 and 3 and shade the interval between them.}$
3. Identify Harder Examples of Regions Defined by Linear Inequalities
• Concept: Regions defined by multiple linear inequalities can be identified by plotting each inequality on the same graph and finding the overlapping region that satisfies all inequalities.
• Example: $\text{For } y \geq 2x + 1 \text{ and } y \leq -x + 4, \text{ plot both lines and shade the region that satisfies both conditions.}$
4. Interpret Solutions Graphically
• Concept: The graphical representation of inequalities can help visually identify the solution set by showing where the graphs of the functions lie above, below, or intersect the relevant lines or curves.
• Example: $\text{For } x^2 – 5x + 6 < 0, \text{ plot the parabola and identify where it is below the x-axis.}$ $\text{For } y \geq 2x + 1 \text{ and } y \leq -x + 4, \text{ find the overlapping shaded region on the graph.}$

#### Common Misconceptions

1. Solve Quadratic Inequalities in One Unknown
• Common Mistake: Incorrectly finding the critical points or misinterpreting the intervals around the roots.
• Example: $\text{Incorrect: For } x^2 – 5x + 6 < 0, \text{ solving the quadratic incorrectly as } (x - 2)(x + 3) \text{ instead of } (x - 2)(x - 3).$ $\text{Correct: Factorise correctly and test intervals properly: }$ $\text{Critical points are } x = 2 \text{ and } x = 3.$ $\text{Test intervals: } \begin{cases} x < 2 \rightarrow (1 - 2)(1 - 3) = 1 \cdot (-2) = -2 \, \text{(negative)} \\ 2 < x < 3 \rightarrow (2.5 - 2)(2.5 - 3) = 0.5 \cdot (-0.5) = -0.25 \, \text{(negative)} \\ x > 3 \rightarrow (4 – 2)(4 – 3) = 2 \cdot 1 = 2 \, \text{(positive)} \end{cases}$ $\text{Solution: } 2 < x < 3.$
2. Represent the Solution Set of a Quadratic Inequality on a Number Line
• Common Mistake: Misrepresenting the solution set on the number line by using closed circles instead of open circles or shading the wrong interval.
• Example: $\text{Incorrect: Representing } 2 < x < 3 \text{ with closed circles at 2 and 3.}$ $\text{Correct: Use open circles at 2 and 3 and shade the interval between them.}$
3. Identify Harder Examples of Regions Defined by Linear Inequalities
• Common Mistake: Misidentifying the region that satisfies all inequalities or incorrectly shading the graph.
• Example: $\text{Incorrect: For } y \geq 2x + 1 \text{ and } y \leq -x + 4, \text{ shading the wrong region or missing the overlapping part.}$ $\text{Correct: Plot both lines accurately and shade the region where } y \geq 2x + 1 \text{ and } y \leq -x + 4 \text{ intersect.}$
4. Interpret Solutions Graphically
• Common Mistake: Misinterpreting the graphical representation by not correctly identifying where the graph is above or below the axis or other lines.
• Example: $\text{Incorrect: For } x^2 – 5x + 6 < 0, \text{ thinking the solution is where the parabola is above the x-axis.}$ $\text{Correct: Identify the interval where the graph is below the x-axis, which represents the solution.}$

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