Solving Quadratic Inequalities - Example 1 - Free Printable

You're solving the inequality:

$$
x^2 + x - 20 < 0
$$

Step 1: Factor the quadratic

The quadratic expression can be factored as:

$$
x^2 + x - 20 = (x + 5)(x - 4)
$$

So the inequality becomes:

$$
(x + 5)(x - 4) < 0
$$

---

Step 2: Find the critical points

Set each factor equal to zero:

- $ x + 5 = 0 $ → $ x = -5 $
- $ x - 4 = 0 $ → $ x = 4 $

These are the critical points that divide the number line into intervals. Note: These values make the expression equal to zero, but since the inequality is strict (< 0), they are not included in the solution.

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Step 3: Use a sign chart (number line)

We test the sign of the expression $ (x + 5)(x - 4) $ in the intervals determined by $ x = -5 $ and $ x = 4 $:

The intervals are:
1. $ (-\infty, -5) $
2. $ (-5, 4) $
3. $ (4, \infty) $

Choose a test point from each interval:

#### Interval 1: $ x < -5 $, say $ x = -6 $
$$
(-6 + 5)(-6 - 4) = (-1)(-10) = 10 > 0 \quad \text{→ Positive}
$$

#### Interval 2: $ -5 < x < 4 $, say $ x = 0 $
$$
(0 + 5)(0 - 4) = (5)(-4) = -20 < 0 \quad \text{→ Negative}
$$

#### Interval 3: $ x > 4 $, say $ x = 5 $
$$
(5 + 5)(5 - 4) = (10)(1) = 10 > 0 \quad \text{→ Positive}
$$

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Step 4: Determine where the expression is negative

We want $ (x + 5)(x - 4) < 0 $, i.e., negative.

That happens only in the interval:

$$
(-5, 4)
$$

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✔ Final Answer:

$$
\boxed{-5 < x < 4}
$$

This is the solution set.

> 💡 Remember: Since the inequality is strict (<), the endpoints $ x = -5 $ and $ x = 4 $ are not included. That’s why we use parentheses, not brackets.

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