The solution of the inequality `(|x+2|-|x|)/(sqrt(8-x^(3))) ge 0` is

Options

A. `[-1,2]`
B. `[1,2]`
C. `[-1,1]`
D. `[0,3sqrt(4))`

Explanation

To solve the inequality \((|x+2|-|x|)/(sqrt{8-x^{3}}) \ge 0\), we will follow these steps:

### Step 1: Determine the domain of the expression
The expression \(\sqrt{8 - x^3}\) must be non-negative, which means:
\[ 8 - x^3 \geq 0 \]
This simplifies to:
\[ x^3 \leq 8 \]
Taking the cube root of both sides gives:
\[ x \leq 2 \]

**Hint:** Remember that the square root function is only defined for non-negative values.

### Step 2: Analyze the numerator
Next, we need to analyze the numerator \(|x + 2| - |x|\). We will consider different cases based on the value of \(x\).

**Case 1:** \(x \geq 0\)
- Here, \(|x| = x\) and \(|x + 2| = x + 2\).
- Thus, the expression becomes:
\[ (x + 2) - x = 2 \]
Since \(2 \geq 0\), this case holds for all \(x \geq 0\).

**Case 2:** \(-2 \leq x < 0\)
- Here, \(|x| = -x\) and \(|x + 2| = x + 2\).
- The expression becomes:
\[ (x + 2) - (-x) = x + 2 + x = 2x + 2 \]
Setting this greater than or equal to zero gives:
\[ 2x + 2 \geq 0 \implies 2x \geq -2 \implies x \geq -1 \]
Thus, in this case, the valid range is \(-1 \leq x < 0\).

**Case 3:** \(x < -2\)
- Here, \(|x| = -x\) and \(|x + 2| = -x - 2\).
- The expression becomes:
\[ (-x - 2) - (-x) = -x - 2 + x = -2 \]
Since \(-2 < 0\), this case does not contribute to the solution.

### Step 3: Combine the results
From the analysis:
- From Case 1, we have \(x \geq 0\).
- From Case 2, we have \(-1 \leq x < 0\).
- From Case 3, there are no valid solutions.

Combining these results, we find that the solution set is:
\[ [-1, 0) \cup [0, 2] \]
This can be simplified to:
\[ [-1, 2) \]

### Step 4: Conclusion
Thus, the solution of the inequality \((|x+2|-|x|)/(sqrt{8-x^{3}}) \ge 0\) is:
\[ x \in [-1, 2) \]

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