The complete solution set of the inequation `sqrt(x+18)le 2-x` is equal to-

Options

A. `[-17,-2]`
B. `(-infty,-2) cup (7,infty)`
C. `(-18,2) cup(7, infty)`
D. `[-18,-2]`

Answer

To solve the inequality \( \sqrt{x + 18} \leq 2 - x \), we will follow these steps:

### Step 1: Identify the domain of the square root
The expression under the square root, \( x + 18 \), must be non-negative:
\[ x + 18 \geq 0 \implies x \geq -18 \]
**Hint:** Ensure that the expression inside the square root is non-negative to avoid complex numbers.

### Step 2: Ensure the right side is non-negative
The right side of the inequality, \( 2 - x \), must also be non-negative:
\[ 2 - x \geq 0 \implies x \leq 2 \]
**Hint:** Check that the right side of the inequality is also non-negative, as square roots are always non-negative.

### Step 3: Square both sides of the inequality
Since both sides are non-negative, we can square the inequality:
\[ \sqrt{x + 18} \leq 2 - x \implies x + 18 \leq (2 - x)^2 \]
Expanding the right side:
\[ x + 18 \leq 4 - 4x + x^2 \]
Rearranging gives:
\[ 0 \leq x^2 - 5x - 14 \]
**Hint:** When squaring both sides, ensure the inequality direction remains the same by confirming both sides are non-negative.

### Step 4: Solve the quadratic inequality
Rearranging the inequality:
\[ x^2 - 5x - 14 \geq 0 \]
Factoring the quadratic:
\[ (x - 7)(x + 2) \geq 0 \]
**Hint:** Factor the quadratic to find the critical points where the expression changes sign.

### Step 5: Determine the intervals
The critical points are \( x = -2 \) and \( x = 7 \). We will test intervals around these points:
- For \( x < -2 \) (e.g., \( x = -3 \)): \( (-3 - 7)(-3 + 2) = (-10)(-1) > 0 \)
- For \( -2 < x < 7 \) (e.g., \( x = 0 \)): \( (0 - 7)(0 + 2) = (-7)(2) < 0 \)
- For \( x > 7 \) (e.g., \( x = 8 \)): \( (8 - 7)(8 + 2) = (1)(10) > 0 \)

Thus, the solution to the inequality is:
\[ x \in (-\infty, -2] \cup [7, \infty) \]
**Hint:** Use test points in each interval to determine where the product is non-negative.

### Step 6: Combine with the domain restrictions
From Step 1 and Step 2, we have the restrictions:
- \( x \geq -18 \)
- \( x \leq 2 \)

Combining these with the intervals found in Step 5, we find the valid solution set:
\[ x \in [-18, -2] \]
**Hint:** Always check the original conditions after finding the solution to ensure they are satisfied.

### Final Answer
The complete solution set of the inequation \( \sqrt{x + 18} \leq 2 - x \) is:
\[ [-18, -2] \]