Quadratic Inequalities - Free Printable

Let's solve the quadratic inequality step by step and verify the solution shown in the image.

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Problem:

Solve the inequality:
$$
x^2 - x \geq 12
$$

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Step 1: Bring all terms to one side

We want to write the inequality in standard form (one side is zero):

$$
x^2 - x - 12 \geq 0
$$

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Step 2: Factor the quadratic expression

We factor $ x^2 - x - 12 $:

We need two numbers that multiply to $-12$ and add to $-1$.
Those numbers are $-4$ and $+3$:

$$
x^2 - x - 12 = (x - 4)(x + 3)
$$

So the inequality becomes:

$$
(x - 4)(x + 3) \geq 0
$$

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Step 3: Find the critical points

Set each factor equal to zero:

- $ x - 4 = 0 $ → $ x = 4 $
- $ x + 3 = 0 $ → $ x = -3 $

These points divide the number line into three intervals:

1. $ (-\infty, -3) $
2. $ (-3, 4) $
3. $ (4, \infty) $

We will test the sign of $ (x - 4)(x + 3) $ in each interval.

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Step 4: Use a sign chart

We analyze the sign of each factor in each interval:

| Interval | $ x + 3 $ | $ x - 4 $ | Product $(x+3)(x-4)$ |
|------------------|-------------|-------------|------------------------|
| $ (-\infty, -3) $ | Negative | Negative | Positive |
| $ (-3, 4) $ | Positive | Negative | Negative |
| $ (4, \infty) $ | Positive | Positive | Positive |

Now, we include the endpoints because the inequality is ≥ 0 (includes equality).

At $ x = -3 $:
$ (x + 3)(x - 4) = (0)(-7) = 0 $ → satisfies $ \geq 0 $

At $ x = 4 $:
$ (x + 3)(x - 4) = (7)(0) = 0 $ → satisfies $ \geq 0 $

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Step 5: Write the solution

The expression is ≥ 0 when:

- $ x \leq -3 $ OR $ x \geq 4 $

In interval notation:

$$
(-\infty, -3] \cup [4, \infty)
$$

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✔ Final Answer:

$$
\boxed{(-\infty, -3] \cup [4, \infty)}
$$

This matches exactly what is shown in the red box in your image.

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Explanation of the Number Line:

- The number line shows:
- A positive sign on $ (-\infty, -3) $
- A negative sign on $ (-3, 4) $
- A positive sign on $ (4, \infty) $
- Red dots at $ x = -3 $ and $ x = 4 $ indicate inclusion (because of ≥).
- Arrows show the direction of the number line.

So, the solution is where the expression is non-negative, i.e., positive or zero — which is correct.

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✔ Conclusion: The solution provided in the image is correct.

Answer: $ \boxed{(-\infty, -3] \cup [4, \infty)} $