The set of all x in `((-pi)/2,pi/2)` satisfying `|4sinx-1| < sqrt(5)` is given by
(a) `(-pi/(10),(3pi)/(10))` (b) `(pi/(10),(3pi)/(10))` (c) `(pi/(10),(3pi)/(10))` (d) none of these

Options

A. `(- pi/10, (3pi)/10)`
B. `(pi/10, (3pi)/10)`
C. `(pi/10, (3pi)/10)`
D. none of these

Explanation

To solve the inequality \( |4\sin x - 1| < \sqrt{5} \) for \( x \) in the interval \( \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \), we can follow these steps:

### Step 1: Remove the Absolute Value
The inequality \( |4\sin x - 1| < \sqrt{5} \) can be split into two separate inequalities:
\[ - \sqrt{5} < 4\sin x - 1 < \sqrt{5} \]

### Step 2: Solve the Two Inequalities
We can solve these inequalities separately.

1. **First Inequality:**
\[ 4\sin x - 1 < \sqrt{5} \]
Adding 1 to both sides:
\[ 4\sin x < \sqrt{5} + 1 \]
Dividing by 4:
\[ \sin x < \frac{\sqrt{5} + 1}{4} \]

2. **Second Inequality:**
\[ -\sqrt{5} < 4\sin x - 1 \]
Adding 1 to both sides:
\[ -\sqrt{5} + 1 < 4\sin x \]
Dividing by 4:
\[ \frac{-\sqrt{5} + 1}{4} < \sin x \]

### Step 3: Evaluate the Bounds
Now we need to evaluate the bounds \( \frac{-\sqrt{5} + 1}{4} \) and \( \frac{\sqrt{5} + 1}{4} \).

1. **Calculate \( \frac{\sqrt{5} + 1}{4} \):**
- The approximate value of \( \sqrt{5} \) is about 2.236, so:
\[ \frac{\sqrt{5} + 1}{4} \approx \frac{2.236 + 1}{4} \approx \frac{3.236}{4} \approx 0.809 \]

2. **Calculate \( \frac{-\sqrt{5} + 1}{4} \):**
\[ \frac{-\sqrt{5} + 1}{4} \approx \frac{-2.236 + 1}{4} \approx \frac{-1.236}{4} \approx -0.309 \]

### Step 4: Set Up the Final Inequality
Now we have:
\[ \frac{-\sqrt{5} + 1}{4} < \sin x < \frac{\sqrt{5} + 1}{4} \]
which translates to:
\[ -0.309 < \sin x < 0.809 \]

### Step 5: Find the Corresponding \( x \) Values
We need to find the values of \( x \) for which \( \sin x \) is in this range.

1. **For \( \sin x = -0.309 \):**
- The angle \( x \) corresponding to \( \sin x = -0.309 \) in the interval \( \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \) is approximately \( -\frac{\pi}{10} \).

2. **For \( \sin x = 0.809 \):**
- The angle \( x \) corresponding to \( \sin x = 0.809 \) is approximately \( \frac{3\pi}{10} \).

### Step 6: Final Interval
Thus, the solution set for \( x \) is:
\[ x \in \left(-\frac{\pi}{10}, \frac{3\pi}{10}\right) \]

### Conclusion
The correct answer is:
**(a) \( \left(-\frac{\pi}{10}, \frac{3\pi}{10}\right) \)**

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