Navigating Quadratic Inequalities

What is a Quadratic Inequality?

A quadratic inequality involves a quadratic expression (an expression with $x^2$) and an inequality symbol (\gt,\ge,\lt,\le) instead of an equals sign. For example: $$x^2 - 3x - 4 \gt 0$$

Your goal is to find the range of $x$-values that satisfy the inequality

 

How is it Different from Solving a Quadratic Equation?

When solving a quadratic equation $(x^2 - 3x - 4 = 0)$, you find the exact values of $x$ that make the equation true. With a quadratic inequality, you find the ranges of $x$ (intervals) where the inequality holds.

 

Methods to Solve Quadratic Inequalities

There are two main methods that can be used to solve quadratic inequalities. These are:

1. Using Number-lines
2. Solve using graphs

 

Steps to Solve Quadratic Inequalities Using Number-lines

1. Write the inequality in standard form: Ensure the inequality is in the form $ax^2 + bx + c < 0$, $ax^2 + bx + c > 0$, $ax^2 + bx + c \leq 0$, or $ax^2 + bx + c \geq 0$

2. Solve the related quadratic equation: Replace the inequality sign with $=$ to solve $ax^2 + bx + c = 0$. This gives the critical values (roots).

3. Draw a number line and test intervals: Use the critical values to divide the number line into intervals. Test a value from each interval in the original inequality to check if the interval satisfies the inequality.

4. Write the solution as an interval: Combine the intervals where the inequality holds true.

 

Steps to Solve Quadratic Inequalities with Graphs

To make quadratic inequalities even clearer, we can visualize them using graphs. A graph helps us see where the quadratic expression is above or below the xxx-axis. Here's how to include and use a graph to solve quadratic inequalities.

1. Graph the Quadratic Function:
   • Plot the quadratic equation $y = ax^2 + bx + c$ on a graph
   • Identify where the graph crosses the $x$-axis (the roots/critical values)

2. Shade the Relevant Region:

   • For $ax^2 + bx + c \gt 0:$ Shade the regions where the graph is above the $x$-axis

 

 

• • For $ax^2 + bx + c \lt 0:$ Shade the regions where the graph is below the $x$-axis

 

 

1. • For inequalities with $\ge$ or $\le$, include the points where the graph touches the $x$-axis

2. Determine the Solution:

   • Use the shaded regions to find the $x$-values that satisfy the inequality

 

Examples

Example 1: Solve $x^2 - 5x + 6 \gt 0$

See the two methods of solving below

Solution using number lines:

1. Write in standard form: The inequality is already in standard form.
2. Solve the related quadratic equation: Solve $x^2 - 5x + 6$: $$(x - 2)(x - 3) = 0  \\ \text{So the critical values are } x =2 \text{ and } x =3$$
3. Test intervals: Divide the number line into three intervals based on $x =2 \text{ and } x =3$:
   • Interval 1: $x \lt 2$
   • Interval 2: $2 \lt x \lt 3$
   • Interval 3: $x \gt 3$ $$\text{Test a value from each interval in } x^2 - 5x + 6 \gt 0:$$
   • For $x = 1$ (Interval 1): $(1)^2 - 5(1) + 6 = 2 (\gt 0, \text{True})$
   • For $x =2.5$ (Interval 2): $(2.5)^2 - 5(2.5) + 6 = -0.25 (\lt 0, \text{False})$
   • For $x = 4$ (Interval 3): $(4)^2 - 5(4) + 6 = 2 (\gt 0, \text{True})$
4. Write the solution: The inequality is true for $x \lt 2$ and $x \gt 3$. The solution is: $$x \in (-\infty,2) \cup (3, \infty)$$

 

Solution using Graphs: 

• Write the inequality in standard form: Already done.
• Solve the related quadratic equation: $$x^2 - 5x + 6 = 0 \\  \text{Factoring gives } (x - 2)(x - 3) = 0, \text{so the roots are } x = 2 \text{ and } x = 3$$
• Plot the graph of $y = x^2 - 5x + 6$ 

• Shade the regions where the graph is above the $x$-axis (these occur when $x \lt 2$ or $x \gt 3$ 

• The solution is: $$x \in (-\infty, 2) \cup (3, \infty) \\ \text{On the graph: }$$
• The parabola opens upwards (as $a \gt 0$)
• The graph is above the $x$-axis to the left of $x = 2$ and to the right of $x = 3$

 

Example 2: Solve $x^2 + x -6 \le 0$

See the two methods of solving below

Solution using number lines:

1. Write in standard form: The inequality is already in standard form.
2. Solve the related quadratic equation: Solve $x^2 + x - 6 = 0$: $$(x + 3)(x - 2) = 0$$ \\ \text{So the critical values are } x = -3 \text{ and } x = 2\]
3. Test intervals: Divide the number line into three intervals: 
   • Interval 1: $x \lt -3$
   • Interval 2: $-3 \le x \le 2$
   • Interval 3: $x \gt 2$ $$\text{ Test a value from each interval in } x^2 + x - 6 \le 0:$$
   • For $x = -4$ (Interval 1): $(-4)^2 + (-4) - 6 = 6 (\gt 0, \text{False})$
   • For $x = 0$ (Interval 2): $(0)^2 + (0) - 6 = -6 (\le 0, \text{ True })$
   • For $x = 3$ (Interval 3): $(3)^2 + (3) - 6 = 6 (\gt 0, \text{ False })$
4. Write the solution: The inequality is true for $-3 \le x \le 2$. The solution is: $$x \in [-3,2]$$

 

Solution using Graphs: 

• Write the inequality in standard form: Already done.
• Solve the related quadratic equation: $$x^2 + x - 6 = 0  \\  \text{Factoring gives } (x + 3)(x - 2) = 0, \text{so the roots are} x = -3 \text{ and } x = 2$$
• Plot the graph of $y = x^2 + x - 6$

• Shade the region below the $x$-axis (where y \le 0)
• The solution is: $$x \in [-3,2]$$