To solve the inequality \( x^2 \leq 12 - x \) and express the solution in interval notation, we can follow these steps:

1. **Rearrange the inequality:**

   \[
   x^2 + x - 12 \leq 0
   \]

2. **Factor the quadratic expression:**

   \[
   x^2 + x - 12 = (x + 4)(x - 3)
   \]

   So, the inequality becomes:

   \[
   (x + 4)(x - 3) \leq 0
   \]

3. **Find the critical points:**

   The critical points are the values of \( x \) for which \( (x + 4)(x - 3) = 0 \). These are:

   \[
   x = -4 \quad \text{and} \quad x = 3
   \]

4. **Test intervals around the critical points:**

   The critical points divide the number line into three intervals: \( (-\infty, -4) \), \( (-4, 3) \), and \( (3, \infty) \).

   - For \( x < -4 \), choose \( x = -5 \):
     \[
     (x + 4)(x - 3) = (-5 + 4)(-5 - 3) = (-1)(-8) = 8 > 0
     \]

   - For \( -4 < x < 3 \), choose \( x = 0 \):
     \[
     (x + 4)(x - 3) = (0 + 4)(0 - 3) = 4 \times (-3) = -12 < 0
     \]

   - For \( x > 3 \), choose \( x = 4 \):
     \[
     (x + 4)(x - 3) = (4 + 4)(4 - 3) = 8 \times 1 = 8 > 0
     \]

5. **Determine the solution set:**

   The inequality \( (x + 4)(x - 3) \leq 0 \) is satisfied in the interval \( [-4, 3] \).

6. **Interval notation:**

   The solution in interval notation is:

   \[
   [-4, 3]
   \]

Would you like a detailed explanation of any of these steps? Here are five related questions to explore further:

1. How can we solve inequalities with quadratic expressions that cannot be factored easily?
2. What is the significance of critical points in solving polynomial inequalities?
3. How do we test which intervals satisfy a given inequality?
4. What are the differences between solving linear and quadratic inequalities?
5. How do we represent the solution of a strict inequality (e.g., \( x^2 < 12 - x \)) in interval notation?

**Tip:** Always verify your factorization or roots calculation by substituting back into the original inequality to ensure accuracy.

Learn how to solve the quadratic inequality x² ≤ 12 − x by factoring and analyzing critical points. The step-by-step solution includes testing intervals and expressing the solution in interval notation. Suitable for students in Grades 9-10.

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