Answer :
To solve the inequality [tex]\(\frac{x+4}{2x-1} < 0\)[/tex], we need to determine where this expression is negative. Let's go through the steps:
1. Identify Critical Points:
- Set the numerator and denominator equal to zero to find the critical points:
- [tex]\(x + 4 = 0\)[/tex] gives [tex]\(x = -4\)[/tex].
- [tex]\(2x - 1 = 0\)[/tex] gives [tex]\(x = \frac{1}{2}\)[/tex].
2. Divide the Number Line:
- The critical points [tex]\(-4\)[/tex] and [tex]\(\frac{1}{2}\)[/tex] divide the number line into three intervals:
- [tex]\((-∞, -4)\)[/tex]
- [tex]\((-4, \frac{1}{2})\)[/tex]
- [tex]\((\frac{1}{2}, ∞)\)[/tex]
3. Test Each Interval:
- Pick a test point from each interval and substitute it into the inequality to check if the expression is negative.
- Interval [tex]\((-∞, -4)\)[/tex]:
- Choose [tex]\(x = -5\)[/tex].
- [tex]\(\frac{-5+4}{2(-5)-1} = \frac{-1}{-11} = \frac{1}{11}\)[/tex] (positive, not negative).
- Interval [tex]\((-4, \frac{1}{2})\)[/tex]:
- Choose [tex]\(x = 0\)[/tex].
- [tex]\(\frac{0+4}{2(0)-1} = \frac{4}{-1} = -4\)[/tex] (negative, satisfies the inequality).
- Interval [tex]\((\frac{1}{2}, ∞)\)[/tex]:
- Choose [tex]\(x = 1\)[/tex].
- [tex]\(\frac{1+4}{2(1)-1} = \frac{5}{1} = 5\)[/tex] (positive, not negative).
4. Conclusion:
- Only the interval [tex]\((-4, \frac{1}{2})\)[/tex] satisfies the inequality [tex]\(\frac{x+4}{2x-1} < 0\)[/tex].
- The correct solution is [tex]\( -4 < x < \frac{1}{2} \)[/tex].
Therefore, the solution is [tex]\(-4 < x < \frac{1}{2}\)[/tex], which corresponds to the choice [tex]\(-4 < x < \frac{1}{2}\)[/tex].
1. Identify Critical Points:
- Set the numerator and denominator equal to zero to find the critical points:
- [tex]\(x + 4 = 0\)[/tex] gives [tex]\(x = -4\)[/tex].
- [tex]\(2x - 1 = 0\)[/tex] gives [tex]\(x = \frac{1}{2}\)[/tex].
2. Divide the Number Line:
- The critical points [tex]\(-4\)[/tex] and [tex]\(\frac{1}{2}\)[/tex] divide the number line into three intervals:
- [tex]\((-∞, -4)\)[/tex]
- [tex]\((-4, \frac{1}{2})\)[/tex]
- [tex]\((\frac{1}{2}, ∞)\)[/tex]
3. Test Each Interval:
- Pick a test point from each interval and substitute it into the inequality to check if the expression is negative.
- Interval [tex]\((-∞, -4)\)[/tex]:
- Choose [tex]\(x = -5\)[/tex].
- [tex]\(\frac{-5+4}{2(-5)-1} = \frac{-1}{-11} = \frac{1}{11}\)[/tex] (positive, not negative).
- Interval [tex]\((-4, \frac{1}{2})\)[/tex]:
- Choose [tex]\(x = 0\)[/tex].
- [tex]\(\frac{0+4}{2(0)-1} = \frac{4}{-1} = -4\)[/tex] (negative, satisfies the inequality).
- Interval [tex]\((\frac{1}{2}, ∞)\)[/tex]:
- Choose [tex]\(x = 1\)[/tex].
- [tex]\(\frac{1+4}{2(1)-1} = \frac{5}{1} = 5\)[/tex] (positive, not negative).
4. Conclusion:
- Only the interval [tex]\((-4, \frac{1}{2})\)[/tex] satisfies the inequality [tex]\(\frac{x+4}{2x-1} < 0\)[/tex].
- The correct solution is [tex]\( -4 < x < \frac{1}{2} \)[/tex].
Therefore, the solution is [tex]\(-4 < x < \frac{1}{2}\)[/tex], which corresponds to the choice [tex]\(-4 < x < \frac{1}{2}\)[/tex].
