Complex Inequalities
Beyond simple inequalities lie quadratic inequalities and graphical regions. These higher-tier skills are essential for modelling real-world constraints, from designing safe structures to finding optimal business solutions. This guide will show you how to solve them with confidence.
Solving Quadratic Inequalities
To solve an inequality like $x^2 – 5x + 6 < 0$, think graphically.
- Find the critical values (roots) by solving $x^2 – 5x + 6 = 0$. Here, $x = 2$ and $x = 3$.
- Sketch the parabola. Since the $x^2$ coefficient is positive, it’s a U-shape crossing the x-axis at the roots.
- Identify the region. The inequality asks for where the graph is $< 0$ (below the x-axis). This is the region between the roots.
- Write the solution: $2 < x < 3$.
Graphing Linear Inequalities
To show a region like $y \ge 2x – 1$ on a graph:
- Draw the boundary line $y = 2x – 1$. Use a solid line for $\ge$ or $\le$, and a dashed line for $>$ or $<$.
- Pick a test point not on the line (e.g., $(0, 0)$).
- Test it in the inequality: $0 \ge 2(0) – 1 \implies 0 \ge -1$. True!
- Shade the correct region. Since $(0, 0)$ satisfied the inequality, shade the side of the line containing the origin.
Worked Examples
Example 1: Quadratic Inequality
Solve $x^2 – 4x – 5 > 0$.
- Critical values: $(x-5)(x+1) = 0 \implies x = 5$ and $x = -1$.
- Sketch: U-shaped parabola crossing at $x = -1$ and $x = 5$.
- Region: We want where the graph is $> 0$ (above the x-axis). This is to the left of $-1$ and to the right of $5$.
Answer: $x < -1$ or $x > 5$
Example 2: Graphical Region
Show the region R satisfying $y \le x + 1$, $y > -2$, and $x < 3$.
- Draw $y = x + 1$ (solid line), shade below.
- Draw $y = -2$ (dashed line), shade above.
- Draw $x = 3$ (dashed line), shade left.
Region R is the triangle where all three shaded areas overlap.
Tutor Insights
🤔 Common Misunderstandings
- Quadratic inequalities: Not sketching the graph! Students often treat it like a linear inequality and miss one part of the solution (e.g., giving only $x > 5$ for $x^2 > 25$).
- Graphical inequalities: Mixing up solid and dashed lines, or shading the wrong region because they didn’t use a test point.
📝 Common Exam Mistakes
- Incorrect critical values due to errors in factorising or the quadratic formula.
- Flipping the sign: Forgetting to reverse the inequality when multiplying or dividing by a negative number.
- Not labelling the region ‘R’ when the question asks for it.
Practice Questions
- Solve $x^2 – 7x + 10 < 0$.
- On a graph, represent the region satisfying $y \le 3x + 1$.
- Solve $2x^2 + 5x – 3 \ge 0$.
- Show on a graph the region R satisfying $y > x – 2$, $y < 4$, and $x \ge 0$.
Show Answers
- Working: $(x-2)(x-5) = 0 \implies x = 2, x = 5$. U-shaped parabola. We want below the x-axis (between the roots).
Answer: $2 < x < 5$. - Draw a solid line for $y = 3x + 1$. Test $(0,0)$: $0 \le 1$ ✓. Shade the region containing the origin (below the line).
- Working: $(2x – 1)(x + 3) = 0 \implies x = 0.5, x = -3$. U-shaped parabola. We want on or above the x-axis (outside the roots).
Answer: $x \le -3$ or $x \ge 0.5$. - Draw a dashed line $y = x – 2$ (shade above); a dashed horizontal line $y = 4$ (shade below); a solid vertical line $x = 0$ (shade to the right). Region R is the triangle where all three areas overlap.
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