https://file.aiquickdraw.com/mathbot/file/1726314316197i6gfwfzh.jpg

The inequality shown in the image is:

\[
32^{3x - 4} > 128^{4x + 3}
\]

To solve this inequality, we can start by expressing both 32 and 128 as powers of 2, since:

\[
32 = 2^5 \quad \text{and} \quad 128 = 2^7
\]

Substituting these into the inequality:

\[
(2^5)^{3x - 4} > (2^7)^{4x + 3}
\]

Using the rule \((a^m)^n = a^{m \cdot n}\), we can simplify both sides:

\[
2^{5(3x - 4)} > 2^{7(4x + 3)}
\]

Simplifying the exponents:

\[
2^{15x - 20} > 2^{28x + 21}
\]

Since the bases are the same, we can compare the exponents:

\[
15x - 20 > 28x + 21
\]

Now, solve this inequality for \(x\):

1. Subtract \(15x\) from both sides:

\[
-20 > 13x + 21
\]

2. Subtract 21 from both sides:

\[
-41 > 13x
\]

3. Divide by 13:

\[
x < \frac{-41}{13}
\]

Thus, the solution is:

\[
x < -\frac{41}{13} \quad \text{or approximately} \quad x < -3.15
\]

Would you like more details on any of the steps, or have any other questions?

Here are 5 related questions:
1. What is the general method to solve inequalities involving exponents?
2. How do you solve inequalities when the bases are different?
3. What are the common rules for simplifying exponent expressions?
4. Can this method be used if the base numbers are not powers of the same number?
5. How would the solution change if the inequality were non-strict, i.e., \( \geq \) instead of \( > \)?

**Tip:** Always check if you can simplify the bases into a common base when dealing with exponent inequalities!

Solve the inequality 32^(3x - 4) > 128^(4x + 3)

Learn how to solve the exponential inequality 32^(3x - 4) > 128^(4x + 3) by expressing the bases as powers of 2 and comparing exponents. A step-by-step solution is provided for students in Grades 10-12.

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