To determine the values of [tex]\( x \)[/tex] for which the expression [tex]\(\sqrt{x+4} \div \sqrt{1-x}\)[/tex] is defined, we need to ensure that the expression inside each square root is non-negative and that we are not dividing by zero.
### Step 1: Analyze [tex]\(\sqrt{x+4}\)[/tex]
For the square root [tex]\(\sqrt{x+4}\)[/tex] to be defined, the expression inside the square root must be non-negative:
[tex]\[ x+4 \geq 0 \][/tex]
Solving this inequality:
[tex]\[ x \geq -4 \][/tex]
### Step 2: Analyze [tex]\(\sqrt{1-x}\)[/tex]
For the square root [tex]\(\sqrt{1-x}\)[/tex] to be defined, the expression inside the square root must be positive:
[tex]\[ 1-x > 0 \][/tex]
Solving this inequality:
[tex]\[ x < 1 \][/tex]
### Step 3: Combine the conditions
We need both conditions to be satisfied simultaneously:
- The condition [tex]\( x \geq -4 \)[/tex]
- The condition [tex]\( x < 1 \)[/tex]
Combining these two conditions, we get:
[tex]\[ -4 \leq x < 1 \][/tex]
### Step 4: Identify the Correct Option
Given the options:
A. [tex]\(4 \leq x \leq 1\)[/tex]
B. [tex]\( 4 > x > 1 \)[/tex]
C. [tex]\(-4 \leq x < 1\)[/tex]
D. [tex]\(4 > x \leq -1\)[/tex]
The correct interval that satisfies [tex]\(-4 \leq x < 1\)[/tex] is:
C. [tex]\(-4 \leq x < 1\)[/tex]
Therefore, the expression [tex]\(\sqrt{x+4} \div \sqrt{1-x}\)[/tex] is defined for [tex]\(\boxed{-4 \leq x < 1}\)[/tex].
