Answer :
To solve the inequality [tex]\(\frac{x+4}{2x-1} < 0\)[/tex], let's follow a step-by-step approach:
1. Understand when a fraction is negative:
- The fraction [tex]\(\frac{A}{B}\)[/tex] is negative when [tex]\(A\)[/tex] and [tex]\(B\)[/tex] have opposite signs.
2. Identify the critical points:
- The critical points occur when the numerator is zero and when the denominator is zero. These points are:
- [tex]\(x + 4 = 0 \rightarrow x = -4\)[/tex]
- [tex]\(2x - 1 = 0 \rightarrow x = \frac{1}{2}\)[/tex]
3. Create intervals based on the critical points:
- The critical points divide the number line into three intervals:
- Interval 1: [tex]\((-\infty, -4)\)[/tex]
- Interval 2: [tex]\((-4, \frac{1}{2})\)[/tex]
- Interval 3: [tex]\((\frac{1}{2}, \infty)\)[/tex]
4. Test each interval to determine where the inequality holds true:
- Choose test points from each interval and substitute them into the inequality [tex]\(\frac{x+4}{2x-1} < 0\)[/tex].
- For Interval 1 [tex]\((-\infty, -4)\)[/tex], choose: [tex]\(x = -5\)[/tex]
[tex]\[
\frac{-5+4}{2(-5)-1} = \frac{-1}{-10-1} = \frac{-1}{-11} = \frac{1}{11} > 0 \quad \text{(inequality does not hold)}
\][/tex]
- For Interval 2 [tex]\((-4, \frac{1}{2})\)[/tex], choose: [tex]\(x = 0\)[/tex]
[tex]\[
\frac{0+4}{2(0)-1} = \frac{4}{-1} = -4 < 0 \quad \text{(inequality holds)}
\][/tex]
- For Interval 3 [tex]\((\frac{1}{2}, \infty)\)[/tex], choose: [tex]\(x = 1\)[/tex]
[tex]\[
\frac{1+4}{2(1)-1} = \frac{5}{1} = 5 > 0 \quad \text{(inequality does not hold)}
\][/tex]
5. Consider the boundary points:
- At [tex]\(x = -4\)[/tex], the fraction is [tex]\(0\)[/tex] which is not less than [tex]\(0\)[/tex].
- At [tex]\(x = \frac{1}{2}\)[/tex], the fraction is undefined since the denominator becomes [tex]\(0\)[/tex].
6. Combine the intervals where the inequality holds true:
- The solution to [tex]\(\frac{x+4}{2x-1} < 0\)[/tex] is the interval where the test was successful but excluding the boundary points where the fraction is zero or undefined.
Hence, the solution is:
[tex]\[
-4 < x < \frac{1}{2}
\][/tex]
From the options given, it corresponds to:
[tex]\[
\boxed{-4 < x < \frac{1}{2}}
\][/tex]
1. Understand when a fraction is negative:
- The fraction [tex]\(\frac{A}{B}\)[/tex] is negative when [tex]\(A\)[/tex] and [tex]\(B\)[/tex] have opposite signs.
2. Identify the critical points:
- The critical points occur when the numerator is zero and when the denominator is zero. These points are:
- [tex]\(x + 4 = 0 \rightarrow x = -4\)[/tex]
- [tex]\(2x - 1 = 0 \rightarrow x = \frac{1}{2}\)[/tex]
3. Create intervals based on the critical points:
- The critical points divide the number line into three intervals:
- Interval 1: [tex]\((-\infty, -4)\)[/tex]
- Interval 2: [tex]\((-4, \frac{1}{2})\)[/tex]
- Interval 3: [tex]\((\frac{1}{2}, \infty)\)[/tex]
4. Test each interval to determine where the inequality holds true:
- Choose test points from each interval and substitute them into the inequality [tex]\(\frac{x+4}{2x-1} < 0\)[/tex].
- For Interval 1 [tex]\((-\infty, -4)\)[/tex], choose: [tex]\(x = -5\)[/tex]
[tex]\[
\frac{-5+4}{2(-5)-1} = \frac{-1}{-10-1} = \frac{-1}{-11} = \frac{1}{11} > 0 \quad \text{(inequality does not hold)}
\][/tex]
- For Interval 2 [tex]\((-4, \frac{1}{2})\)[/tex], choose: [tex]\(x = 0\)[/tex]
[tex]\[
\frac{0+4}{2(0)-1} = \frac{4}{-1} = -4 < 0 \quad \text{(inequality holds)}
\][/tex]
- For Interval 3 [tex]\((\frac{1}{2}, \infty)\)[/tex], choose: [tex]\(x = 1\)[/tex]
[tex]\[
\frac{1+4}{2(1)-1} = \frac{5}{1} = 5 > 0 \quad \text{(inequality does not hold)}
\][/tex]
5. Consider the boundary points:
- At [tex]\(x = -4\)[/tex], the fraction is [tex]\(0\)[/tex] which is not less than [tex]\(0\)[/tex].
- At [tex]\(x = \frac{1}{2}\)[/tex], the fraction is undefined since the denominator becomes [tex]\(0\)[/tex].
6. Combine the intervals where the inequality holds true:
- The solution to [tex]\(\frac{x+4}{2x-1} < 0\)[/tex] is the interval where the test was successful but excluding the boundary points where the fraction is zero or undefined.
Hence, the solution is:
[tex]\[
-4 < x < \frac{1}{2}
\][/tex]
From the options given, it corresponds to:
[tex]\[
\boxed{-4 < x < \frac{1}{2}}
\][/tex]
