https://file.aiquickdraw.com/mathbot/file/1726575652229zwudijfq.jpg

The inequality shown in the image is:

\[
\frac{8x^3}{27} - 1 \leq 0
\]

Let's solve this step by step:

### Step 1: Isolate the fraction
\[
\frac{8x^3}{27} \leq 1
\]

### Step 2: Multiply both sides by 27
\[
8x^3 \leq 27
\]

### Step 3: Divide by 8
\[
x^3 \leq \frac{27}{8}
\]

### Step 4: Take the cube root of both sides
\[
x \leq \sqrt[3]{\frac{27}{8}} = \frac{3}{2}
\]

Thus, the solution is:

\[
x \leq \frac{3}{2}
\]

This inequality shows that \(x\) can take any value less than or equal to \( \frac{3}{2} \).

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

Here are 5 related questions to expand this topic:
1. How do you solve inequalities involving higher-degree polynomials?
2. What is the general method to solve fractional inequalities?
3. How does taking cube roots differ from square roots in inequalities?
4. Can you explain the concept of domain restrictions when solving inequalities?
5. How would you graph the solution set for this inequality on a number line?

**Tip:** When solving inequalities involving powers, be mindful of the properties of even and odd roots. For odd powers, the inequality direction does not change when taking roots.

Determine the solution of the inequality shown in the image.

This guide explains how to solve the cubic inequality 8x^3/27 - 1 <= 0, with a step-by-step breakdown of isolating terms, simplifying, and taking cube roots. This inequality resolution is key for students learning algebraic manipulation and properties of roots. Suitable for high school students studying advanced algebra topics.

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