The inequality you provided is:

\[
36q^2 \leq 32
\]

### Step 1: Isolate \( q^2 \)

First, divide both sides of the inequality by 36:

\[
q^2 \leq \frac{32}{36}
\]

Simplify the fraction:

\[
q^2 \leq \frac{8}{9}
\]

### Step 2: Solve for \( q \)

To solve for \( q \), take the square root of both sides. When taking the square root of both sides in an inequality, you consider both the positive and negative square roots:

\[
-\sqrt{\frac{8}{9}} \leq q \leq \sqrt{\frac{8}{9}}
\]

Simplifying the square roots:

\[
-\frac{2\sqrt{2}}{3} \leq q \leq \frac{2\sqrt{2}}{3}
\]

### Step 3: Express the solution in interval notation

The solution set for \( q \) is:

\[
\left[ -\frac{2\sqrt{2}}{3}, \frac{2\sqrt{2}}{3} \right]
\]

This is the interval where \( q \) satisfies the inequality \( 36q^2 \leq 32 \).

Would you like further details, or do you have any questions?

### 5 Related Questions:
1. How do you solve quadratic inequalities?
2. Why do we consider both positive and negative square roots when solving inequalities?
3. How do you graph the solution of a quadratic inequality?
4. What are the rules for solving inequalities involving squares?
5. How does interval notation represent inclusive and exclusive bounds?

### Tip:
When taking the square root of both sides of an inequality, always remember to account for both the positive and negative roots!

Learn how to solve the inequality 36q^2 <= 32 with a detailed step-by-step solution. This problem covers key concepts including quadratic inequalities and square roots. Suitable for high school students in Grades 9-12.

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