Answer :
To solve the inequality [tex]\(\frac{x+4}{2x-1}<0\)[/tex], let's break it down step-by-step:
1. Identify Critical Points:
- The expression changes its sign at points where either the numerator or the denominator is zero.
- Set the numerator to zero: [tex]\(x + 4 = 0\)[/tex] which gives [tex]\(x = -4\)[/tex].
- Set the denominator to zero: [tex]\(2x - 1 = 0\)[/tex] which gives [tex]\(x = \frac{1}{2}\)[/tex].
2. Consider Intervals:
- These critical points divide the number line into intervals: [tex]\((-\infty, -4)\)[/tex], [tex]\((-4, \frac{1}{2})\)[/tex], [tex]\((\frac{1}{2}, \infty)\)[/tex].
3. Test Each Interval:
- Interval [tex]\((-\infty, -4)\)[/tex]:
- Choose a test point like [tex]\(x = -5\)[/tex].
- Substitute into the inequality: [tex]\(\frac{-5 + 4}{2(-5) - 1} = \frac{-1}{-11} = \frac{1}{11} > 0\)[/tex].
- This interval is positive.
- Interval [tex]\((-4, \frac{1}{2})\)[/tex]:
- Choose a test point like [tex]\(x = 0\)[/tex].
- Substitute into the inequality: [tex]\(\frac{0 + 4}{2(0) - 1} = \frac{4}{-1} = -4 < 0\)[/tex].
- This interval is negative.
- Interval [tex]\((\frac{1}{2}, \infty)\)[/tex]:
- Choose a test point like [tex]\(x = 1\)[/tex].
- Substitute into the inequality: [tex]\(\frac{1 + 4}{2(1) - 1} = \frac{5}{1} = 5 > 0\)[/tex].
- This interval is positive.
4. Consider Endpoint Inclusion:
- At [tex]\(x = -4\)[/tex], the expression is [tex]\(\frac{-4 + 4}{2(-4) - 1} = \frac{0}{-9} = 0\)[/tex]. The inequality strictly requires the expression to be less than zero, not equal to zero.
- At [tex]\(x = \frac{1}{2}\)[/tex], the denominator becomes zero, making the expression undefined.
5. Conclusion:
- The solution to the inequality is [tex]\(-4 < x < \frac{1}{2}\)[/tex].
The set of [tex]\(x\)[/tex] where the inequality holds true is [tex]\((-4, \frac{1}{2})\)[/tex].
1. Identify Critical Points:
- The expression changes its sign at points where either the numerator or the denominator is zero.
- Set the numerator to zero: [tex]\(x + 4 = 0\)[/tex] which gives [tex]\(x = -4\)[/tex].
- Set the denominator to zero: [tex]\(2x - 1 = 0\)[/tex] which gives [tex]\(x = \frac{1}{2}\)[/tex].
2. Consider Intervals:
- These critical points divide the number line into intervals: [tex]\((-\infty, -4)\)[/tex], [tex]\((-4, \frac{1}{2})\)[/tex], [tex]\((\frac{1}{2}, \infty)\)[/tex].
3. Test Each Interval:
- Interval [tex]\((-\infty, -4)\)[/tex]:
- Choose a test point like [tex]\(x = -5\)[/tex].
- Substitute into the inequality: [tex]\(\frac{-5 + 4}{2(-5) - 1} = \frac{-1}{-11} = \frac{1}{11} > 0\)[/tex].
- This interval is positive.
- Interval [tex]\((-4, \frac{1}{2})\)[/tex]:
- Choose a test point like [tex]\(x = 0\)[/tex].
- Substitute into the inequality: [tex]\(\frac{0 + 4}{2(0) - 1} = \frac{4}{-1} = -4 < 0\)[/tex].
- This interval is negative.
- Interval [tex]\((\frac{1}{2}, \infty)\)[/tex]:
- Choose a test point like [tex]\(x = 1\)[/tex].
- Substitute into the inequality: [tex]\(\frac{1 + 4}{2(1) - 1} = \frac{5}{1} = 5 > 0\)[/tex].
- This interval is positive.
4. Consider Endpoint Inclusion:
- At [tex]\(x = -4\)[/tex], the expression is [tex]\(\frac{-4 + 4}{2(-4) - 1} = \frac{0}{-9} = 0\)[/tex]. The inequality strictly requires the expression to be less than zero, not equal to zero.
- At [tex]\(x = \frac{1}{2}\)[/tex], the denominator becomes zero, making the expression undefined.
5. Conclusion:
- The solution to the inequality is [tex]\(-4 < x < \frac{1}{2}\)[/tex].
The set of [tex]\(x\)[/tex] where the inequality holds true is [tex]\((-4, \frac{1}{2})\)[/tex].
