Solution set of `x-sqrt(1-|x|)lt0`, is

Options

A. `[-1,(-1+sqrt(5))/2)`
B. `[-1,1]`
C. `[-1,(-1+sqrt(5))/2]`
D. `(-1,(-1+sqrt(5))/2)`

Explanation

To solve the inequality \( x - \sqrt{1 - |x|} < 0 \), we can follow these steps:

### Step 1: Rearranging the Inequality
We start by rearranging the inequality:
\[ x < \sqrt{1 - |x|} \]

### Step 2: Squaring Both Sides
Next, we square both sides to eliminate the square root. However, we must be cautious about the implications of squaring both sides, as it can introduce extraneous solutions:
\[ x^2 < 1 - |x| \]

### Step 3: Rearranging Again
Rearranging gives us:
\[ x^2 + |x| - 1 < 0 \]

### Step 4: Considering Cases for |x|
Since we have the absolute value, we need to consider two cases: \( x \geq 0 \) and \( x < 0 \).

#### Case 1: \( x \geq 0 \)
In this case, \( |x| = x \):
\[ x^2 + x - 1 < 0 \]
Now, we can find the roots of the equation \( x^2 + x - 1 = 0 \) using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 1 \cdot (-1)}}{2 \cdot 1} = \frac{-1 \pm \sqrt{5}}{2} \]
The roots are:
\[ x_1 = \frac{-1 + \sqrt{5}}{2}, \quad x_2 = \frac{-1 - \sqrt{5}}{2} \]
Since we are in the case where \( x \geq 0 \), we only consider \( x_1 = \frac{-1 + \sqrt{5}}{2} \).

#### Case 2: \( x < 0 \)
In this case, \( |x| = -x \):
\[ x^2 - x - 1 < 0 \]
Again, we find the roots:
\[ x = \frac{1 \pm \sqrt{1^2 - 4 \cdot 1 \cdot (-1)}}{2 \cdot 1} = \frac{1 \pm \sqrt{5}}{2} \]
The roots are:
\[ x_1 = \frac{1 + \sqrt{5}}{2}, \quad x_2 = \frac{1 - \sqrt{5}}{2} \]
Since we are in the case where \( x < 0 \), we only consider \( x_2 = \frac{1 - \sqrt{5}}{2} \).

### Step 5: Finding the Intervals
Now we need to determine the intervals where the expressions \( x^2 + x - 1 < 0 \) and \( x^2 - x - 1 < 0 \) hold true.

1. For \( x^2 + x - 1 < 0 \):
- The roots are \( \frac{-1 - \sqrt{5}}{2} \) and \( \frac{-1 + \sqrt{5}}{2} \).
- The interval where this expression is negative is:
\[ \left( \frac{-1 - \sqrt{5}}{2}, \frac{-1 + \sqrt{5}}{2} \right) \]

2. For \( x^2 - x - 1 < 0 \):
- The roots are \( \frac{1 - \sqrt{5}}{2} \) and \( \frac{1 + \sqrt{5}}{2} \).
- The interval where this expression is negative is:
\[ \left( \frac{1 - \sqrt{5}}{2}, \frac{1 + \sqrt{5}}{2} \right) \]

### Step 6: Combining the Results
Now we combine the results from both cases:
- For \( x < 0 \), we take the interval from the first case.
- For \( x \geq 0 \), we take the interval from the second case.

### Final Solution Set
The solution set for the inequality \( x - \sqrt{1 - |x|} < 0 \) is:
\[ \left[ -1, \frac{-1 + \sqrt{5}}{2} \right) \]