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The given inequality is:

\[
3x^2 - 8 \leq -2x
\]

To solve this inequality by graphing, let's first rearrange it into a standard form:

\[
3x^2 + 2x - 8 \leq 0
\]

Now, we treat the expression \( 3x^2 + 2x - 8 = 0 \) as a quadratic equation, and we will solve for \(x\) by finding the roots using the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

where \( a = 3 \), \( b = 2 \), and \( c = -8 \).

Let me calculate that for you.The roots of the quadratic equation \( 3x^2 + 2x - 8 = 0 \) are approximately \( x = 1.33 \) and \( x = -2.00 \).

Now, to solve the inequality \( 3x^2 + 2x - 8 \leq 0 \), we can consider the intervals determined by the roots: 

- \( (-\infty, -2] \)
- \( [-2, 1.33] \)
- \( [1.33, \infty) \)

Since the quadratic is a parabola that opens upwards (because the coefficient of \( x^2 \) is positive), the expression \( 3x^2 + 2x - 8 \leq 0 \) is satisfied between the roots, meaning the solution is:

\[
-2 \leq x \leq 1.33
\]

Thus, the solution to the inequality is \( x \in [-2, 1.33] \).

Would you like more details on graphing or any further clarification?

Here are five related questions:
1. How can we check solutions for quadratic inequalities using test points?
2. Why does the quadratic expression change signs at the roots?
3. How does the quadratic formula help solve inequalities?
4. What are the key features of the graph of a quadratic function?
5. How would the solution change if the inequality was strict (i.e., \(<\) instead of \( \leq \))?

**Tip:** When solving quadratic inequalities, the roots divide the number line into intervals, and testing values within those intervals helps determine where the inequality holds.

Solve the inequality by graphing. Round your answers to the nearest hundredth. 3x^2 - 8 ≤ -2x

Learn how to solve the inequality 3x^2 - 8 ≤ -2x by graphing. This problem covers key concepts such as quadratic equations, inequalities, and graphing, and is suitable for students in Grades 9-11. Step-by-step solutions included.

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