Welcome to Quadratic Inequalities!
Welcome to one of the most rewarding and visual topics in CCEA GCSE Further Mathematics: Quadratic Inequalities. Don't worry if inequalities have felt confusing in the past. In this chapter, you will discover a reliable, visual method that turns these problems into simple sketches you can solve with confidence every single time!
Why does this matter? In real life, engineers, economists, and scientists rarely deal with exact equations. Instead, they work with safety limits, profit margins, and tolerance ranges. For example, a business might need to know: "For what range of ticket prices is our profit greater than zero?" That is a quadratic inequality in action.
Quick Review: What is an Inequality?
An inequality compares two values or expressions. The symbols you need to know are:
• \(<\) means less than
• \(>\) means greater than
• \(\le\) means less than or equal to
• \(\ge\) means greater than or equal to
Did you know? While a standard quadratic equation like \(x^2 - 5x + 6 = 0\) has just two specific answers (\(x = 2\) and \(x = 3\)), a quadratic inequality has an entire range (set) of values as its answer!
The Golden Rule of Quadratic Inequalities
Never try to solve a quadratic inequality purely using algebra like a linear inequality. For example, if you have \(x^2 > 9\), it is tempting to take the square root and write \(x > 3\). But this misses all the negative numbers like \(x = -5\) (since \((-5)^2 = 25 > 9\))!
To get every question right, we always use the Sketch Method.
The 4-Step Method to Success
Follow these four simple steps for every quadratic inequality:
Step 1: Rearrange into standard form
Rearrange your inequality so that all terms are on one side and \(0\) is on the other side. Make sure the coefficient of \(x^2\) is positive (a standard "happy face" \(\cup\)-shaped parabola):
\(ax^2 + bx + c > 0\), \(ax^2 + bx + c < 0\), \(ax^2 + bx + c \ge 0\), or \(ax^2 + bx + c \le 0\).
Step 2: Find the "Critical Values"
Temporarily replace the inequality sign with an equals sign: \(ax^2 + bx + c = 0\). Solve this quadratic equation by factorising (or using the quadratic formula) to find the boundary points where the graph crosses the \(x\)-axis. These numbers are called your critical values.
Step 3: Sketch the curve
Draw a simple \(x\)-axis line and sketch a \(\cup\)-shaped parabola crossing through your critical values (smaller number on the left, larger on the right).
Step 4: Identify and write the solution region
Look at what the inequality asks for:
• If it asks for \(< 0\) or \(\le 0\), look for the part of the curve below the \(x\)-axis.
• If it asks for \(> 0\) or \(\ge 0\), look for the parts of the curve above the \(x\)-axis.
Understanding the Two Types of Solutions
Depending on whether you are looking above or below the axis, your final answer will always take one of two structures:
1. The "Between" Region (Below the Axis)
When the curve is below the \(x\)-axis (for \(< 0\) or \(\le 0\)), the graph is trapped between the two critical values, say \(p\) and \(q\) (where \(p < q\)).
This gives a single combined inequality:
\(p < x < q\) (or \(p \le x \le q\))
Memory aid: Think of this as the "sandwich" — the values of \(x\) are sandwiched inside the two boundaries.
2. The "Outside" Regions (Above the Axis)
When the curve is above the \(x\)-axis (for \(> 0\) or \(\ge 0\)), the graph goes off to the far left and the far right.
This gives two separate inequalities joined by the word or:
\(x < p \text{ or } x > q\) (or \(x \le p \text{ or } x \ge q\))
Memory aid: Think of these as the "wings" — two separate wings stretching outwards away from the middle.
Worked Examples
Example 1: Solving a "Less Than or Equal To" Inequality
Question: Solve the inequality \(x^2 - 5x + 6 \le 0\).
Solution:
Step 1: The inequality is already in standard form with zero on the right side: \(x^2 - 5x + 6 \le 0\).
Step 2: Find the critical values by solving \(x^2 - 5x + 6 = 0\):
\((x - 2)(x - 3) = 0\)
Critical values are \(x = 2\) and \(x = 3\).
Step 3: Sketch a \(\cup\)-shaped parabola crossing the \(x\)-axis at \(x = 2\) and \(x = 3\).
Step 4: We want where the expression is \(\le 0\) (on or below the \(x\)-axis). Looking at the sketch, the curve dips below the axis between \(2\) and \(3\).
Final Answer: \(2 \le x \le 3\)
Example 2: Rearranging and Solving a "Greater Than" Inequality
Question: Find the set of values of \(x\) for which \(2x^2 + 5x > 3\).
Solution:
Step 1: Rearrange to get \(0\) on the right-hand side:
\(2x^2 + 5x - 3 > 0\)
Step 2: Solve the equation \(2x^2 + 5x - 3 = 0\):
\((2x - 1)(x + 3) = 0\)
Critical values are \(x = \frac{1}{2}\) and \(x = -3\).
Step 3: Sketch a \(\cup\)-shaped parabola passing through \(-3\) (on the left) and \(\frac{1}{2}\) (on the right).
Step 4: We want where the expression is \(> 0\) (strictly above the \(x\)-axis). This occurs to the left of \(-3\) and to the right of \(\frac{1}{2}\).
Final Answer: \(x < -3 \text{ or } x > \frac{1}{2}\)
Example 3: Dealing with a Negative \(x^2\) Term
Question: Solve \(5 - 4x - x^2 \ge 0\).
Solution:
It is always easiest to work with a positive \(x^2\) term. Let's move everything to the right-hand side (or multiply the entire inequality by \(-1\) and flip the sign):
\(0 \ge x^2 + 4x - 5\)
Which means: \(x^2 + 4x - 5 \le 0\)
Step 2: Find the critical values:
\(x^2 + 4x - 5 = 0\)
\((x + 5)(x - 1) = 0\)
Critical values are \(x = -5\) and \(x = 1\).
Step 3: Sketch the \(\cup\)-shaped parabola crossing at \(-5\) and \(1\).
Step 4: We want where \(x^2 + 4x - 5 \le 0\) (below or on the axis).
Final Answer: \(-5 \le x \le 1\)
Common Mistakes to Avoid in Exams
1. Writing two separate regions as a single impossible statement:
Never write \(1 < x < -5\) or \(3 < x < 2\). This makes no mathematical sense! If the solution consists of two separate outside regions, always write them separately with the word "or", like \(x < -5 \text{ or } x > 1\).
2. Forgetting to flip the inequality sign when multiplying/dividing by a negative:
If you multiply or divide both sides of an inequality by a negative number, always reverse the inequality symbol (e.g., \(>\) becomes \(<\)).
3. Mixing up strict and non-strict inequality signs:
If the question uses \(<\) or \(>\), your answer must use \(<\) or \(>\). If the question uses \(\le\) or \(\ge\), your answer must use \(\le\) or \(\ge\).
4. Dividing by \(x\):
Never divide an inequality by \(x\) (e.g., turning \(x^2 > 4x\) into \(x > 4\)). Because \(x\) could be negative, dividing by it might flip the inequality sign, and you will also lose critical boundary values!
Key Takeaways Summary
• Always rearrange your quadratic inequality so one side is zero before doing anything else.
• Find the critical values by setting the expression equal to zero and solving for \(x\).
• Always sketch a quick \(\cup\)-shaped parabola to visually check your answer.
• For \(< 0\) (below axis): write a single combined range, e.g., \(p < x < q\).
• For \(> 0\) (above axis): write two separate statements, e.g., \(x < p \text{ or } x > q\).