https://file.aiquickdraw.com/mathbot/file/172898643562524ot0p69.jpg

The image presents three quadratic inequalities that need to be solved. Let's go through each one step by step.

### 1. \( x^2 + 5x - 24 > 0 \)
We first solve the corresponding equation:
\[
x^2 + 5x - 24 = 0
\]
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) with \( a = 1 \), \( b = 5 \), and \( c = -24 \):
\[
x = \frac{-5 \pm \sqrt{5^2 - 4(1)(-24)}}{2(1)}
\]
\[
x = \frac{-5 \pm \sqrt{25 + 96}}{2} = \frac{-5 \pm \sqrt{121}}{2} = \frac{-5 \pm 11}{2}
\]
So, the solutions are:
\[
x = \frac{-5 + 11}{2} = 3 \quad \text{and} \quad x = \frac{-5 - 11}{2} = -8
\]
These are the critical points where the parabola touches the x-axis. To solve the inequality \( x^2 + 5x - 24 > 0 \), we need to determine where the parabola is above the x-axis.

The parabola opens upwards because the coefficient of \( x^2 \) is positive. Therefore, the solution to \( x^2 + 5x - 24 > 0 \) is when \( x < -8 \) or \( x > 3 \).

**Solution:**
\[
(-\infty, -8) \cup (3, \infty)
\]

---

### 2. \( x^2 + 6x \leq 0 \)
We solve the corresponding equation:
\[
x^2 + 6x = 0
\]
Factor the equation:
\[
x(x + 6) = 0
\]
So, \( x = 0 \) or \( x = -6 \). These are the critical points.

Next, we analyze the sign of the quadratic expression \( x^2 + 6x \) in the intervals defined by \( x = -6 \) and \( x = 0 \):
- For \( x < -6 \), both factors \( x \) and \( x + 6 \) are negative, so \( x^2 + 6x > 0 \).
- For \( -6 < x < 0 \), \( x \) is negative and \( x + 6 \) is positive, so \( x^2 + 6x < 0 \).
- For \( x > 0 \), both factors are positive, so \( x^2 + 6x > 0 \).

Thus, \( x^2 + 6x \leq 0 \) holds in the interval \( [-6, 0] \).

**Solution:**
\[
[-6, 0]
\]

---

### 3. \( 2x^2 - 72 \geq 0 \)
First, simplify the inequality:
\[
2x^2 - 72 \geq 0 \quad \Rightarrow \quad x^2 \geq 36
\]
This inequality implies:
\[
x \geq 6 \quad \text{or} \quad x \leq -6
\]
So the solution set is \( x \leq -6 \) or \( x \geq 6 \).

**Solution:**
\[
(-\infty, -6] \cup [6, \infty]
\]

---

Do you want further details on any of these solutions?

Here are 5 related questions to explore further:
1. How do you solve quadratic equations using factoring instead of the quadratic formula?
2. Can you graph the solutions to these inequalities to better visualize them?
3. What is the difference between strict and non-strict inequalities?
4. How do you determine whether a parabola opens upwards or downwards?
5. How do you check solutions for quadratic inequalities using test points?

**Tip:** For quadratic inequalities, always check the critical points (where the expression equals zero) and test intervals between those points to determine the sign of the expression.

Find the solution set of the following quadratic inequality. 1) x^2 + 5x - 24 > 0, 2) x^2 + 6x ≤ 0, 3) 2x^2 - 72 ≥ 0.

Step-by-step solutions to quadratic inequalities x^2 + 5x - 24 > 0, x^2 + 6x ≤ 0, and 2x^2 - 72 ≥ 0 using the quadratic formula and factoring. Learn key algebraic concepts for Grades 9-12.

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