Answer :
To solve the expression [tex]\(\left(\frac{x+3}{2x+7}\right)^{\frac{1}{2}}\)[/tex], we need to understand what it's asking us to do. This expression is a rational function under a square root. Let’s go through the steps to simplify or analyze it.
### Step-by-Step Solution:
1. Identify the rational expression within the square root:
[tex]\[ \frac{x + 3}{2x + 7} \][/tex]
2. Recognize that the expression needs to be non-negative under the square root:
For the square root to be real, the entire expression inside it must be non-negative, i.e., [tex]\(\frac{x+3}{2x+7} \geq 0\)[/tex].
3. Determine the critical points for the rational expression:
These are the points where the numerator and the denominator are zero.
- Numerator (x + 3 = 0):
[tex]\[ x = -3 \][/tex]
- Denominator (2x + 7 = 0):
[tex]\[ 2x + 7 = 0 \implies x = -\frac{7}{2} \][/tex]
4. Determine the intervals to test the sign of the rational function across different ranges of [tex]\(x\)[/tex]:
The critical points divide the number line into the following intervals:
[tex]\[ (-\infty, -\frac{7}{2}), \quad (-\frac{7}{2}, -3), \quad (-3, \infty) \][/tex]
5. Test the sign of the rational function in each interval:
- Interval [tex]\((-∞, -\frac{7}{2})\)[/tex]:
- Pick a point, say [tex]\(x = -4\)[/tex]:
[tex]\[ \frac{-4 + 3}{2(-4) + 7} = \frac{-1}{-8 + 7} = \frac{-1}{-1} = 1 \quad (\text{positive}) \][/tex]
- Interval [tex]\((- \frac{7}{2}, -3)\)[/tex]:
- Pick a point, say [tex]\(x = -4\)[/tex]:
[tex]\[ \frac{-4 + 3}{2(-4) + 7} = \frac{-1}{-8 + 7} = \frac{-1}{-1} = 1 \quad (\text{positive}) \][/tex]
- Interval [tex]\((-3, ∞)\)[/tex]:
- Pick a point, say [tex]\(x = 0\)[/tex]:
[tex]\[ \frac{0 + 3}{2(0) + 7} = \frac{3}{7} \quad (\text{positive}) \][/tex]
6. Compile all positive intervals and critical points.
### Decision:
- The square root function is valid and real where [tex]\(\frac{x + 3}{2x + 7} \geq 0\)[/tex].
- Based on our interval testing, the expression is defined for [tex]\(x \in (-\infty, -\frac{7}{2}) \cup [-3, \infty)\)[/tex].
### Simplified Expression:
Therefore, for the given intervals, the expression [tex]\(\left(\frac{x+3}{2x+7}\right)^{\frac{1}{2}}\)[/tex] can be simplified to [tex]\( \sqrt{\frac{x+3}{2x+7}}\)[/tex].
### Conclusion:
Thus, the solution to the given mathematical expression is:
[tex]\[ \boxed{\sqrt{\frac{x+3}{2x+7}}} \][/tex]
And it is defined in the intervals [tex]\( x\in (-\infty, -\frac{7}{2}) \cup [-3, \infty)\)[/tex].
### Step-by-Step Solution:
1. Identify the rational expression within the square root:
[tex]\[ \frac{x + 3}{2x + 7} \][/tex]
2. Recognize that the expression needs to be non-negative under the square root:
For the square root to be real, the entire expression inside it must be non-negative, i.e., [tex]\(\frac{x+3}{2x+7} \geq 0\)[/tex].
3. Determine the critical points for the rational expression:
These are the points where the numerator and the denominator are zero.
- Numerator (x + 3 = 0):
[tex]\[ x = -3 \][/tex]
- Denominator (2x + 7 = 0):
[tex]\[ 2x + 7 = 0 \implies x = -\frac{7}{2} \][/tex]
4. Determine the intervals to test the sign of the rational function across different ranges of [tex]\(x\)[/tex]:
The critical points divide the number line into the following intervals:
[tex]\[ (-\infty, -\frac{7}{2}), \quad (-\frac{7}{2}, -3), \quad (-3, \infty) \][/tex]
5. Test the sign of the rational function in each interval:
- Interval [tex]\((-∞, -\frac{7}{2})\)[/tex]:
- Pick a point, say [tex]\(x = -4\)[/tex]:
[tex]\[ \frac{-4 + 3}{2(-4) + 7} = \frac{-1}{-8 + 7} = \frac{-1}{-1} = 1 \quad (\text{positive}) \][/tex]
- Interval [tex]\((- \frac{7}{2}, -3)\)[/tex]:
- Pick a point, say [tex]\(x = -4\)[/tex]:
[tex]\[ \frac{-4 + 3}{2(-4) + 7} = \frac{-1}{-8 + 7} = \frac{-1}{-1} = 1 \quad (\text{positive}) \][/tex]
- Interval [tex]\((-3, ∞)\)[/tex]:
- Pick a point, say [tex]\(x = 0\)[/tex]:
[tex]\[ \frac{0 + 3}{2(0) + 7} = \frac{3}{7} \quad (\text{positive}) \][/tex]
6. Compile all positive intervals and critical points.
### Decision:
- The square root function is valid and real where [tex]\(\frac{x + 3}{2x + 7} \geq 0\)[/tex].
- Based on our interval testing, the expression is defined for [tex]\(x \in (-\infty, -\frac{7}{2}) \cup [-3, \infty)\)[/tex].
### Simplified Expression:
Therefore, for the given intervals, the expression [tex]\(\left(\frac{x+3}{2x+7}\right)^{\frac{1}{2}}\)[/tex] can be simplified to [tex]\( \sqrt{\frac{x+3}{2x+7}}\)[/tex].
### Conclusion:
Thus, the solution to the given mathematical expression is:
[tex]\[ \boxed{\sqrt{\frac{x+3}{2x+7}}} \][/tex]
And it is defined in the intervals [tex]\( x\in (-\infty, -\frac{7}{2}) \cup [-3, \infty)\)[/tex].
