Answer :
To solve the inequality [tex]\(\frac{x+4}{2x-1} < 0\)[/tex], we need to analyze where this expression changes sign and where it fulfills the condition of being less than zero.
### Step 1: Determine Critical Points
Critical points occur where the expression is zero or undefined.
1. Zero Numerator: Solve [tex]\(x + 4 = 0\)[/tex].
- This gives [tex]\(x = -4\)[/tex].
2. Zero Denominator: Solve [tex]\(2x - 1 = 0\)[/tex].
- This gives [tex]\(x = \frac{1}{2}\)[/tex].
### Step 2: Test Sign Changes
Next, we test intervals around these critical points to see where the inequality is negative.
- Interval [tex]\((-∞, -4)\)[/tex]:
- Pick a test value, for example [tex]\(x = -5\)[/tex].
- [tex]\(\frac{-5 + 4}{2(-5) - 1} = \frac{-1}{-10 - 1} = \frac{-1}{-11} = \frac{1}{11}\)[/tex] which is positive.
- Interval [tex]\((-4, \frac{1}{2})\)[/tex]:
- Pick a test value, for example [tex]\(x = 0\)[/tex].
- [tex]\(\frac{0 + 4}{2(0) - 1} = \frac{4}{-1} = -4\)[/tex] which is negative.
- Interval [tex]\((\frac{1}{2}, ∞)\)[/tex]:
- Pick a test value, for example [tex]\(x = 1\)[/tex].
- [tex]\(\frac{1 + 4}{2(1) - 1} = \frac{5}{1} = 5\)[/tex] which is positive.
### Step 3: Solution Set
From the sign tests, the only interval where the expression is negative is [tex]\((-4, \frac{1}{2})\)[/tex].
### Step 4: Consider Open and Closed Intervals
- At [tex]\(x = -4\)[/tex]: The expression [tex]\(\frac{x+4}{2x-1}\)[/tex] turns zero and we need it to be strictly less than zero.
- At [tex]\(x = \frac{1}{2}\)[/tex]: The inequality is undefined since the denominator becomes zero.
So the solution is [tex]\(-4 < x < \frac{1}{2}\)[/tex].
Therefore, the correct solution to the inequality [tex]\(\frac{x + 4}{2x - 1} < 0\)[/tex] is:
[tex]\[
-4 < x < \frac{1}{2}
\][/tex]
### Step 1: Determine Critical Points
Critical points occur where the expression is zero or undefined.
1. Zero Numerator: Solve [tex]\(x + 4 = 0\)[/tex].
- This gives [tex]\(x = -4\)[/tex].
2. Zero Denominator: Solve [tex]\(2x - 1 = 0\)[/tex].
- This gives [tex]\(x = \frac{1}{2}\)[/tex].
### Step 2: Test Sign Changes
Next, we test intervals around these critical points to see where the inequality is negative.
- Interval [tex]\((-∞, -4)\)[/tex]:
- Pick a test value, for example [tex]\(x = -5\)[/tex].
- [tex]\(\frac{-5 + 4}{2(-5) - 1} = \frac{-1}{-10 - 1} = \frac{-1}{-11} = \frac{1}{11}\)[/tex] which is positive.
- Interval [tex]\((-4, \frac{1}{2})\)[/tex]:
- Pick a test value, for example [tex]\(x = 0\)[/tex].
- [tex]\(\frac{0 + 4}{2(0) - 1} = \frac{4}{-1} = -4\)[/tex] which is negative.
- Interval [tex]\((\frac{1}{2}, ∞)\)[/tex]:
- Pick a test value, for example [tex]\(x = 1\)[/tex].
- [tex]\(\frac{1 + 4}{2(1) - 1} = \frac{5}{1} = 5\)[/tex] which is positive.
### Step 3: Solution Set
From the sign tests, the only interval where the expression is negative is [tex]\((-4, \frac{1}{2})\)[/tex].
### Step 4: Consider Open and Closed Intervals
- At [tex]\(x = -4\)[/tex]: The expression [tex]\(\frac{x+4}{2x-1}\)[/tex] turns zero and we need it to be strictly less than zero.
- At [tex]\(x = \frac{1}{2}\)[/tex]: The inequality is undefined since the denominator becomes zero.
So the solution is [tex]\(-4 < x < \frac{1}{2}\)[/tex].
Therefore, the correct solution to the inequality [tex]\(\frac{x + 4}{2x - 1} < 0\)[/tex] is:
[tex]\[
-4 < x < \frac{1}{2}
\][/tex]
