If `-1 le [2x^2 -3] lt 2`, then x belongs to

Options

A. `-sqrt((5)/(2)) lt x le -1` only
B. ` le x lt sqrt((5)/(2))` only
C. `- sqrt((5)/(2)) lr x le -1` or `1 le x le sqrt((5)/(2))`
D. `-1 le x le 1`

Answer

To solve the inequality \(-1 \leq 2x^2 - 3 < 2\), we will break it down into two parts and solve each part step by step.

### Step 1: Solve the left part of the inequality
We start with the left part of the inequality:
\[ 2x^2 - 3 \geq -1 \]
Add 3 to both sides:
\[ 2x^2 \geq 2 \]
Now, divide both sides by 2:
\[ x^2 \geq 1 \]
Taking the square root of both sides gives:
\[ x \leq -1 \quad \text{or} \quad x \geq 1 \]

### Step 2: Solve the right part of the inequality
Now, we will solve the right part of the inequality:
\[ 2x^2 - 3 < 2 \]
Add 3 to both sides:
\[ 2x^2 < 5 \]
Now, divide both sides by 2:
\[ x^2 < \frac{5}{2} \]
Taking the square root of both sides gives:
\[ -\sqrt{\frac{5}{2}} < x < \sqrt{\frac{5}{2}} \]

### Step 3: Combine the results
Now we need to combine the results from both parts:
1. From the left part, we have \(x \leq -1\) or \(x \geq 1\).
2. From the right part, we have \(-\sqrt{\frac{5}{2}} < x < \sqrt{\frac{5}{2}}\).

Now we will find the intersection of these two results.

- For \(x \leq -1\):
- The intersection with \(-\sqrt{\frac{5}{2}} < x\) gives us \(-\sqrt{\frac{5}{2}} < x \leq -1\).

- For \(x \geq 1\):
- The intersection with \(x < \sqrt{\frac{5}{2}}\) gives us \(1 \leq x < \sqrt{\frac{5}{2}}\).

### Final Answer
Thus, the solution set is:
\[ x \in \left(-\sqrt{\frac{5}{2}}, -1\right] \cup \left[1, \sqrt{\frac{5}{2}}\right) \]