If `log_(2) | 4 - 5x| lt 2,` then x `in`

Options

A. `(0, (5)/(4) ) `
B. `(0, (4)/(5))`
C. `(0, (5)/(8))`
D. `((4)/(5),(8)/(5))`

Answer

To solve the inequality \( \log_2 |4 - 5x| < 2 \), we will follow these steps:

### Step 1: Rewrite the logarithmic inequality
The inequality \( \log_2 |4 - 5x| < 2 \) can be rewritten using the property of logarithms. This means that:
\[ |4 - 5x| < 2^2 \]
Thus, we have:
\[ |4 - 5x| < 4 \]

### Step 2: Remove the absolute value
The absolute value inequality \( |A| < B \) can be expressed as:
\[ -B < A < B \]
Applying this to our inequality gives:
\[ -4 < 4 - 5x < 4 \]

### Step 3: Solve the left part of the inequality
Starting with the left part:
\[ -4 < 4 - 5x \]
Subtract 4 from both sides:
\[ -8 < -5x \]
Now, divide by -5 (remember to flip the inequality sign):
\[ \frac{8}{5} > x \quad \text{or} \quad x < \frac{8}{5} \]

### Step 4: Solve the right part of the inequality
Now, for the right part:
\[ 4 - 5x < 4 \]
Subtract 4 from both sides:
\[ -5x < 0 \]
Now, divide by -5 (again, flip the inequality sign):
\[ x > 0 \]

### Step 5: Combine the results
From the two parts, we have:
\[ 0 < x < \frac{8}{5} \]

### Final Answer
Thus, the solution for \( x \) is:
\[ x \in (0, \frac{8}{5}) \]

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