Answer :
To solve the inequality [tex]\(\frac{x+4}{2x-1} < 0\)[/tex], we need to determine when the expression is negative. Let's go through the steps:
1. Identify critical points:
- The critical points occur when the numerator [tex]\(x + 4 = 0\)[/tex] and the denominator [tex]\(2x - 1 = 0\)[/tex].
- Solve [tex]\(x + 4 = 0\)[/tex] to find [tex]\(x = -4\)[/tex].
- Solve [tex]\(2x - 1 = 0\)[/tex] to find [tex]\(x = \frac{1}{2}\)[/tex].
2. Determine intervals:
- These critical points divide the number line into intervals: [tex]\(x < -4\)[/tex], [tex]\(-4 < x < \frac{1}{2}\)[/tex], and [tex]\(x > \frac{1}{2}\)[/tex].
3. Test each interval:
- For each interval, pick a test point to determine if the expression is positive or negative.
- Interval [tex]\(x < -4\)[/tex]:
- Pick a test point [tex]\(x = -5\)[/tex].
- Substitute into the inequality [tex]\(\frac{x+4}{2x-1} = \frac{-5+4}{2(-5)-1} = \frac{-1}{-11}\)[/tex], which is positive.
- Interval [tex]\(-4 < x < \frac{1}{2}\)[/tex]:
- Pick a test point [tex]\(x = 0\)[/tex].
- Substitute into the inequality [tex]\(\frac{x+4}{2x-1} = \frac{0+4}{2(0)-1} = \frac{4}{-1}\)[/tex], which is negative.
- Interval [tex]\(x > \frac{1}{2}\)[/tex]:
- Pick a test point [tex]\(x = 1\)[/tex].
- Substitute into the inequality [tex]\(\frac{x+4}{2x-1} = \frac{1+4}{2(1)-1} = \frac{5}{1}\)[/tex], which is positive.
4. Conclusion:
- The inequality is satisfied (negative) in the interval [tex]\(-4 < x < \frac{1}{2}\)[/tex].
Thus, the solution for the inequality [tex]\(\frac{x+4}{2x-1} < 0\)[/tex] is:
[tex]\[ -4 < x < \frac{1}{2} \][/tex]
1. Identify critical points:
- The critical points occur when the numerator [tex]\(x + 4 = 0\)[/tex] and the denominator [tex]\(2x - 1 = 0\)[/tex].
- Solve [tex]\(x + 4 = 0\)[/tex] to find [tex]\(x = -4\)[/tex].
- Solve [tex]\(2x - 1 = 0\)[/tex] to find [tex]\(x = \frac{1}{2}\)[/tex].
2. Determine intervals:
- These critical points divide the number line into intervals: [tex]\(x < -4\)[/tex], [tex]\(-4 < x < \frac{1}{2}\)[/tex], and [tex]\(x > \frac{1}{2}\)[/tex].
3. Test each interval:
- For each interval, pick a test point to determine if the expression is positive or negative.
- Interval [tex]\(x < -4\)[/tex]:
- Pick a test point [tex]\(x = -5\)[/tex].
- Substitute into the inequality [tex]\(\frac{x+4}{2x-1} = \frac{-5+4}{2(-5)-1} = \frac{-1}{-11}\)[/tex], which is positive.
- Interval [tex]\(-4 < x < \frac{1}{2}\)[/tex]:
- Pick a test point [tex]\(x = 0\)[/tex].
- Substitute into the inequality [tex]\(\frac{x+4}{2x-1} = \frac{0+4}{2(0)-1} = \frac{4}{-1}\)[/tex], which is negative.
- Interval [tex]\(x > \frac{1}{2}\)[/tex]:
- Pick a test point [tex]\(x = 1\)[/tex].
- Substitute into the inequality [tex]\(\frac{x+4}{2x-1} = \frac{1+4}{2(1)-1} = \frac{5}{1}\)[/tex], which is positive.
4. Conclusion:
- The inequality is satisfied (negative) in the interval [tex]\(-4 < x < \frac{1}{2}\)[/tex].
Thus, the solution for the inequality [tex]\(\frac{x+4}{2x-1} < 0\)[/tex] is:
[tex]\[ -4 < x < \frac{1}{2} \][/tex]
