The set of all x in `(-pi,pi)` satisfying `|4 sin x-1| lt sqrt(5)` is given by

Options

A. `((-9pi)/10 , (7pi)/10)`
B. `((-pi)/10 , (3pi)/10)`
C. `((-2pi)/10 , (3pi)/10)`
D. `(-pi , pi)`

Answer

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

### Step 1: Remove the Absolute Value
The expression \( |4 \sin x - 1| < \sqrt{5} \) implies two inequalities:
\[ -\sqrt{5} < 4 \sin x - 1 < \sqrt{5} \]

### Step 2: Solve the Inequalities
We can break this into two parts:

1. \( 4 \sin x - 1 < \sqrt{5} \)
2. \( 4 \sin x - 1 > -\sqrt{5} \)

#### For the first inequality:
\[ 4 \sin x - 1 < \sqrt{5} \]
Adding 1 to both sides:
\[ 4 \sin x < \sqrt{5} + 1 \]
Dividing by 4:
\[ \sin x < \frac{\sqrt{5} + 1}{4} \]

#### For the second inequality:
\[ 4 \sin x - 1 > -\sqrt{5} \]
Adding 1 to both sides:
\[ 4 \sin x > -\sqrt{5} + 1 \]
Dividing by 4:
\[ \sin x > \frac{-\sqrt{5} + 1}{4} \]

### Step 3: Find the Values of \( \sin x \)
Now we need to evaluate the bounds:
- Let \( a = \frac{\sqrt{5} + 1}{4} \)
- Let \( b = \frac{-\sqrt{5} + 1}{4} \)

### Step 4: Calculate \( a \) and \( b \)
Calculating \( a \):
\[ a = \frac{\sqrt{5} + 1}{4} \approx \frac{2.236 + 1}{4} \approx \frac{3.236}{4} \approx 0.809 \]

Calculating \( b \):
\[ b = \frac{-\sqrt{5} + 1}{4} \approx \frac{-2.236 + 1}{4} \approx \frac{-1.236}{4} \approx -0.309 \]

### Step 5: Solve for \( x \)
Now we need to find \( x \) such that:
\[ -0.309 < \sin x < 0.809 \]

### Step 6: Determine the Angles
Using the unit circle:
1. For \( \sin x = 0.809 \):
- The angles in \( (-\pi, \pi) \) are approximately \( x \approx \frac{7\pi}{10} \) and \( x \approx -\frac{3\pi}{10} \).

2. For \( \sin x = -0.309 \):
- The angles in \( (-\pi, \pi) \) are approximately \( x \approx -\frac{9\pi}{10} \) and \( x \approx \frac{3\pi}{10} \).

### Step 7: Combine the Results
The solution set for \( x \) is:
\[ x \in \left(-\frac{9\pi}{10}, \frac{7\pi}{10}\right) \]

### Final Answer
The set of all \( x \) in \( (-\pi, \pi) \) satisfying \( |4 \sin x - 1| < \sqrt{5} \) is:
\[ \boxed{\left(-\frac{9\pi}{10}, \frac{7\pi}{10}\right)} \]