How many integer values of `x` satisfy the inequality `(32)/(243) lt ((2)/(3))^(x^(2)) lt (9)/(4) ((8)/(27))^(x)`

Options

A. `3`
B. `4`
C. `0`
D. `2`

Explanation

To solve the inequality

\[ \frac{32}{243} < \left(\frac{2}{3}\right)^{x^2} < \frac{9}{4} \left(\frac{8}{27}\right)^{x} \]

we will first express all terms in terms of \(\frac{2}{3}\).

### Step 1: Rewrite the terms in the inequality

1. **Rewrite \(\frac{32}{243}\)**:
\[ \frac{32}{243} = \frac{2^5}{3^5} = \left(\frac{2}{3}\right)^5 \]

2. **Rewrite \(\frac{9}{4}\)**:
\[ \frac{9}{4} = \frac{3^2}{2^2} = \left(\frac{3}{2}\right)^2 = \left(\frac{2}{3}\right)^{-2} \]

3. **Rewrite \(\frac{8}{27}\)**:
\[ \frac{8}{27} = \frac{2^3}{3^3} = \left(\frac{2}{3}\right)^3 \]

Now, substituting these back into the inequality gives:

\[ \left(\frac{2}{3}\right)^5 < \left(\frac{2}{3}\right)^{x^2} < \left(\frac{2}{3}\right)^{-2} \left(\frac{2}{3}\right)^{3x} \]

This simplifies to:

\[ \left(\frac{2}{3}\right)^5 < \left(\frac{2}{3}\right)^{x^2} < \left(\frac{2}{3}\right)^{3x - 2} \]

### Step 2: Analyze the inequalities

Since \(\frac{2}{3} < 1\), we can reverse the inequalities when taking logarithms or comparing powers:

1. From \(\left(\frac{2}{3}\right)^5 < \left(\frac{2}{3}\right)^{x^2}\):
\[ x^2 < 5 \]
This implies:
\[ -\sqrt{5} < x < \sqrt{5} \]

2. From \(\left(\frac{2}{3}\right)^{x^2} < \left(\frac{2}{3}\right)^{3x - 2}\):
\[ x^2 > 3x - 2 \]
Rearranging gives:
\[ x^2 - 3x + 2 > 0 \]
Factoring:
\[ (x - 1)(x - 2) > 0 \]

### Step 3: Solve the quadratic inequality

The critical points are \(x = 1\) and \(x = 2\). We analyze the sign of the product:

- For \(x < 1\), both factors are negative, so the product is positive.
- For \(1 < x < 2\), one factor is negative and the other is positive, so the product is negative.
- For \(x > 2\), both factors are positive, so the product is positive.

Thus, the solution to the inequality is:
\[ x < 1 \quad \text{or} \quad x > 2 \]

### Step 4: Combine the results

We have two inequalities:
1. \(-\sqrt{5} < x < \sqrt{5}\)
2. \(x < 1\) or \(x > 2\)

The approximate values of \(\sqrt{5}\) are about \(2.236\). Therefore, the first inequality gives us:
\[ -2.236 < x < 2.236 \]

Now we combine:
- From \(x < 1\) and \(-2.236 < x < 2.236\), we have:
\[ -2.236 < x < 1 \]

- From \(x > 2\) and \(-2.236 < x < 2.236\), we have:
\[ \text{No solution since } x > 2 \text{ is outside the range.} \]

### Step 5: Find integer solutions

The integer solutions in the interval \(-2.236 < x < 1\) are:
\[ -2, -1, 0 \]

Thus, the integer values of \(x\) that satisfy the inequality are \(-2, -1, 0\).

### Final Answer
The total number of integer values of \(x\) that satisfy the inequality is **3**.

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