Let's solve the inequality $4 - 9y^2 \geq 0$.

Step 1: Rearrange the inequality

We want to isolate the terms involving $y$. Start by moving the $4$ to the other side of the inequality:

$4 - 9y^2 \geq 0$

Subtract 4 from both sides:

$-9y^2 \geq -4$

Step 2: Eliminate the negative sign

To simplify the inequality, we can divide both sides by $-9$. Remember that dividing by a negative number reverses the inequality direction:

$y^2 \leq \frac{4}{9}$

Step 3: Solve for $y$

To find the values of $y$, take the square root of both sides. Since we are dealing with an inequality, remember that taking the square root introduces both positive and negative solutions:

$-\frac{2}{3} \leq y \leq \frac{2}{3}$

Final solution:

The solution to the inequality is:

$-\frac{2}{3} \leq y \leq \frac{2}{3}$

This means that $y$ can take any value between $-\frac{2}{3}$ and $\frac{2}{3}$, inclusive.

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Would you like further explanation on any part of this process?

Here are some related questions you might find helpful:

1. How to solve quadratic inequalities in general?
2. What happens if the inequality is strict, i.e., $4 - 9y^2 > 0$?
3. How do we graph the solution to this inequality?
4. How do you solve inequalities with higher powers of $y$?
5. How can this process be applied to other polynomial inequalities?

Tip: When solving inequalities, always pay attention to whether you’re multiplying or dividing by negative numbers, as this reverses the direction of the inequality.