To solve the inequality [tex]\(\frac{x+4}{2x-1} < 0\)[/tex], we'll proceed step-by-step.
1. Identify Critical Points:
The expression [tex]\(\frac{x+4}{2x-1} < 0\)[/tex] changes sign at the points where the numerator ([tex]\(x+4\)[/tex]) or the denominator ([tex]\(2x-1\)[/tex]) equals zero. These points are:
- [tex]\(x + 4 = 0 \Rightarrow x = -4\)[/tex]
- [tex]\(2x - 1 = 0 \Rightarrow x = \frac{1}{2}\)[/tex]
2. Determine the Intervals:
Using the critical points, we can divide the number line into intervals to test the sign of the expression [tex]\(\frac{x+4}{2x-1}\)[/tex]:
- Interval 1: [tex]\( (-\infty, -4) \)[/tex]
- Interval 2: [tex]\( (-4, \frac{1}{2}) \)[/tex]
- Interval 3: [tex]\( (\frac{1}{2}, \infty) \)[/tex]
3. Test a Point in Each Interval:
We test a point in each interval to see where the inequality [tex]\(\frac{x+4}{2x-1} < 0\)[/tex] holds true:
- For Interval 1: Pick [tex]\(x = -5\)[/tex]
[tex]\[
\frac{-5 + 4}{2(-5) - 1} = \frac{-1}{-11} = \frac{1}{11} > 0 \quad \text{(Not valid)}
\][/tex]
- For Interval 2: Pick [tex]\(x = 0\)[/tex]
[tex]\[
\frac{0 + 4}{2(0) - 1} = \frac{4}{-1} = -4 < 0 \quad \text{(Valid)}
\][/tex]
- For Interval 3: Pick [tex]\(x = 1\)[/tex]
[tex]\[
\frac{1 + 4}{2(1) - 1} = \frac{5}{1} = 5 > 0 \quad \text{(Not valid)}
\][/tex]
4. Combine the Intervals and Consider the Critical Points:
The valid interval where our inequality [tex]\(\frac{x+4}{2x-1} < 0\)[/tex] holds is [tex]\((-4, \frac{1}{2})\)[/tex]. Note the inequality is strict (i.e., [tex]\(\)[/tex] < 0\)), hence we do not include the endpoints [tex]\(x = -4\)[/tex] and [tex]\(x = \frac{1}{2}\)[/tex].
So, the solution set for the inequality [tex]\(\frac{x+4}{2x-1} < 0\)[/tex] is:
[tex]\[
-4 < x < \frac{1}{2}
\][/tex]
Therefore, the correct answer is: [tex]\(\boxed{-4 < x < \frac{1}{2}}\)[/tex].
