Quadratic Inequalities and Set Notation

Understanding Regions and Set Notation

Once you have identified the correct parts of your sketch, you must state the solution set. This requires specific mathematical formatting known as set notation, which is wrapped in curly brackets: $\{ \}$.

For a standard U-shaped parabola, there are two possible types of regions:

• Continuous Regions (The Valley): When the inequality is $< 0$ or $\le 0$, the curve dips below the x-axis between the two critical values. This forms a single, continuous region (e.g., $-2 < x < 5$). In set notation, this is written as $\{x : -2 < x < 5\}$. The colon means "such that".
• Disjoint Regions (The Tails): When the inequality is $> 0$ or $\ge 0$, the curve sits above the x-axis on the outside edges of the critical values. Because these regions do not touch, they cannot be written as a single inequality. We use the union symbol ($\cup$) to combine them (e.g., $\{x : x < -2\} \cup \{x : x > 5\}$).
• Intersection: While the intersection symbol ($\cap$) mathematically represents the overlap of sets, OCR exams consistently prefer the continuous compound inequality format ($-2 < x < 5$) to describe overlapping regions.

Representing Solutions on a Number Line

Inequalities can also be plotted visually on a number line using circles and arrows.

• An open circle ($\circ$) is used for strict inequalities ($<$ and $>$), showing that the critical value is not included.
• A closed circle ($\bullet$) is used for inclusive inequalities ($\le$ and $\ge$), showing that the critical value is included.
• For continuous regions, draw a solid horizontal line connecting the two circles.
• For disjoint regions, draw outward-pointing arrows moving away from each circle.

Worked Example: Disjoint Regions (Factorisable)

Solve the inequality $x^2 - 2x \ge 15$ and express your answer using set notation.

Step 1: Rearrange to zero.

• $x^2 - 2x - 15 \ge 0$

Step 2: Find the critical values by solving $x^2 - 2x - 15 = 0$.

• $(x - 5)(x + 3) = 0$
• Critical values are $x = 5$ and $x = -3$.

Step 3: Sketch the parabola.

• Draw a U-shaped curve crossing the x-axis at $-3$ and $5$.

Step 4: Identify the region and provide the final inequality range.

• The rearranged inequality is $\ge 0$, so we want the regions above or on the x-axis ("the tails").
• The x-values must be $x \le -3$ or $x \ge 5$.

Step 5: Express in set notation.

• $\{x : x \le -3\} \cup \{x : x \ge 5\}$

Worked Example: Continuous Region (Non-Factorisable)

Find the set of values for which $x^2 + 6x + 4 < 0$. Give your answer in set notation, leaving values in exact surd form.

Step 1: Rearrange to zero.

• The inequality is already in standard form: $x^2 + 6x + 4 < 0$.

Step 2: Find critical values. It does not factorise easily, so use the quadratic formula.

• $a = 1$, $b = 6$, $c = 4$

• $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

• $x = \frac{-6 \pm \sqrt{6^2 - 4(1)(4)}}{2(1)}$

• $x = \frac{-6 \pm \sqrt{36 - 16}}{2}$

• $x = \frac{-6 \pm \sqrt{20}}{2}$

• $x = \frac{-6 \pm 2\sqrt{5}}{2}$

• Critical values are $x = -3 + \sqrt{5}$ and $x = -3 - \sqrt{5}$.

Step 3: Sketch the parabola.

• Draw a U-shaped curve crossing the x-axis at $-3 - \sqrt{5}$ and $-3 + \sqrt{5}$.

Step 4: Identify the region and state the inequality range.

• The inequality is $< 0$, so we want the region below the x-axis (the "valley" between the roots).
• $-3 - \sqrt{5} < x < -3 + \sqrt{5}$

Step 5: Express in set notation.

• $\{x : -3 - \sqrt{5} < x < -3 + \sqrt{5}\}$