Quadratic Inequalities Revision Notes for Grade 12 NSC Matric Mathematics

Quadratic Inequalities

Understanding quadratic inequalities

A quadratic inequality is a mathematical statement involving a quadratic expression with an inequality symbol. These expressions take the form:

$ax^2 + bx + c > 0$
$ax^2 + bx + c < 0$
$ax^2 + bx + c ≥ 0$
$ax^2 + bx + c ≤ 0$

where $a$ , $b$ , and $c$ are constants, and a ≠ 0 (this ensures we have a quadratic expression).

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The solution to a quadratic inequality is a range of x-values that make the inequality true, rather than specific individual values. This is what makes quadratic inequalities different from quadratic equations.

Method 1: Number line approach

This systematic method involves five clear steps to find the solution intervals using critical points and test intervals.

Step 1: Rewrite in standard form

Make sure your inequality is in the form $ax^2 + bx + c > 0$ or $ax^2 + bx + c < 0$ . If the coefficient of $x^2$ is negative, multiply the entire inequality by $-1$ , remembering to reverse the inequality sign.

Step 2: Find the critical points

Solve the equation $ax^2 + bx + c = 0$ to find the critical values (also called roots or x-intercepts). You can use:

Factorisation (if the expression factorises easily)
Quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
Completing the square method

Step 3: Draw a number line

Mark your critical values on a number line. These points divide the number line into different intervals.

Step 4: Test intervals

Choose a test value from each interval created by the critical points. Substitute these values into the original quadratic expression to determine whether the result is positive or negative. Mark each interval with + (positive) or - (negative).

Step 5: Write the solution

For inequalities with $> 0$ or $≥ 0$ , choose intervals marked with $+$
For inequalities with $< 0$ or $≤ 0$ , choose intervals marked with $-$
Use proper inequality notation for your final answer
Remember: $<$ and $>$ exclude the critical points, while $≤$ and $≥$ include them

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Critical Concept: When multiplying or dividing an inequality by a negative number, you must flip the inequality sign. This is one of the most common mistakes in quadratic inequality problems.

Worked example 1: Number line method

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Worked Example: Solving x² < 25 using the number line method

Solve for x if $x^2 < 25$

Step 1: Express in standard form

$x^2 - 25 < 0$

Factorise the expression: $(x - 5)(x + 5) < 0$

Step 2: Find critical values

Set each factor equal to zero:

$x - 5 = 0 → x = 5$
$x + 5 = 0 → x = -5$

Step 3: Mark critical points

Place $-5$ and $5$ on the number line.

Step 4: Test intervals

Choose test points in the three regions:

$x < -5$ : Let $x = -10$ $(-10)^2 - 25 = 100 - 25 = 75$ (positive)
$-5 < x < 5$ : Let $x = 0$
$(0)^2 - 25 = -25$ (negative)
$x > 5$ : Let $x = 9$ $(9)^2 - 25 = 81 - 25 = 56$ (positive)

Step 5: Determine the solution

The quadratic expression is negative between $-5$ and $5$ . Since we need $\< 0$ (strictly less than), the critical points are not included.

Final solution: $-5 < x < 5$

Method 2: Parabola sketch approach

This visual method uses the graph of the quadratic function to identify solution intervals. The parabola sketch approach provides an intuitive way to understand where the quadratic expression is positive or negative.

Step 1: Rewrite in standard form

Ensure your inequality is in the form $ax^2 + bx + c = 0$ .

Step 2: Find the critical values

These are the x-intercepts where the parabola crosses the x-axis.

Step 3: Sketch a rough parabola

If a > 0, the parabola opens upward
If $a < 0$ , the parabola opens downward
Mark where the parabola is above or below the x-axis

Step 4: Determine the solution

Select the x-values where the parabola satisfies the inequality condition.

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Key tip for parabola method:

For $> 0$ or $≥ 0$ : Select intervals where the graph is above the x-axis
For $< 0$ or $≤ 0$ : Select intervals where the graph is below the x-axis

Worked example 2: Parabola method

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Worked Example: Solving -x² ≤ 3x - 3 using the parabola method

Solve for x if $-x^2 ≤ 3x - 3$

Step 1: Express in standard form

Rearrange to get 0 on one side: $-x^2 - 3x + 3 ≤ 0$

Multiply by $-1$ (flip inequality sign): $x^2 + 3x - 3 ≥ 0$

Step 2: Find critical values

Use the quadratic formula with $a = 1$ , $b = 3$ , $c = -3$ :

$x = \frac{-3 \pm \sqrt{(3)^2 - 4(1)(-3)}}{2(1)}$

$x = \frac{-3 \pm \sqrt{9 + 12}}{2}$

$x = \frac{-3 \pm \sqrt{21}}{2}$

Critical values: $x = \frac{-3 - \sqrt{21}}{2}$ and $x = \frac{-3 + \sqrt{21}}{2}$

Step 3: Sketch the parabola

Since $a = 1 > 0$ , the parabola opens upward. The graph is above the x-axis for $x ≤ \frac{-3 - \sqrt{21}}{2}$ or $x ≥ \frac{-3 + \sqrt{21}}{2}$ .

Final solution: $x ≤ \frac{-3 - \sqrt{21}}{2}$ or $x ≥ \frac{-3 + \sqrt{21}}{2}$

Exam tips

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Essential Exam Tips:

Always check your inequality direction when multiplying or dividing by negative numbers
Be careful with strict vs non-strict inequalities - this affects whether you include the critical points
Test your answer by substituting a value from your solution back into the original inequality
Show clear working for finding critical points, especially when using the quadratic formula
Draw neat number lines with clearly marked critical points and interval signs

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Key Points to Remember:

Quadratic inequalities can be solved using factorisation, the quadratic formula, or by sketching parabolas
Use a number line to systematically test intervals and determine solution regions
Multiplying by a negative reverses the inequality sign - this is a common exam trap
Parabola sketches provide a visual way to identify where the expression is positive or negative
Always express your final answer using proper inequality notation with correct inclusion/exclusion of boundary points

Quadratic Inequalities (Grade 12 NSC Matric Mathematics): Revision Notes