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Answer :
Let's solve the inequality [tex]\(\frac{x+4}{2x-1} < 0\)[/tex].
### Step 1: Find the critical points.
The critical points occur where the expression is equal to zero or undefined.
1. Numerator equal to zero: The numerator is [tex]\(x + 4\)[/tex]. Set it equal to zero:
[tex]\[
x + 4 = 0 \implies x = -4
\][/tex]
2. Denominator equal to zero: The denominator is [tex]\(2x - 1\)[/tex]. Set it equal to zero:
[tex]\[
2x - 1 = 0 \implies x = \frac{1}{2}
\][/tex]
### Step 2: Test intervals.
These critical points divide the number line into intervals. We will test each interval to see where the inequality holds true. The intervals are:
- [tex]\(x < -4\)[/tex]
- [tex]\(-4 < x < \frac{1}{2}\)[/tex]
- [tex]\(x > \frac{1}{2}\)[/tex]
Test the intervals:
1. Interval [tex]\(x < -4\)[/tex]: Choose a test point like [tex]\(x = -5\)[/tex].
[tex]\[
\frac{x+4}{2x-1} = \frac{-5+4}{2(-5)-1} = \frac{-1}{-11} = \frac{1}{11} > 0
\][/tex]
This interval does not satisfy the inequality.
2. Interval [tex]\(-4 < x < \frac{1}{2}\)[/tex]: Choose a test point like [tex]\(x = 0\)[/tex].
[tex]\[
\frac{x+4}{2x-1} = \frac{0+4}{2(0)-1} = \frac{4}{-1} = -4 < 0
\][/tex]
This interval satisfies the inequality.
3. Interval [tex]\(x > \frac{1}{2}\)[/tex]: Choose a test point like [tex]\(x = 1\)[/tex].
[tex]\[
\frac{x+4}{2x-1} = \frac{1+4}{2(1)-1} = \frac{5}{1} = 5 > 0
\][/tex]
This interval does not satisfy the inequality.
### Step 3: Formulate the solution.
The solution is the interval where the inequality is true. From our testing, this is:
[tex]\(-4 < x < \frac{1}{2}\)[/tex]
Thus, the solution is [tex]\(-4 < x < \frac{1}{2}\)[/tex].
### Step 1: Find the critical points.
The critical points occur where the expression is equal to zero or undefined.
1. Numerator equal to zero: The numerator is [tex]\(x + 4\)[/tex]. Set it equal to zero:
[tex]\[
x + 4 = 0 \implies x = -4
\][/tex]
2. Denominator equal to zero: The denominator is [tex]\(2x - 1\)[/tex]. Set it equal to zero:
[tex]\[
2x - 1 = 0 \implies x = \frac{1}{2}
\][/tex]
### Step 2: Test intervals.
These critical points divide the number line into intervals. We will test each interval to see where the inequality holds true. The intervals are:
- [tex]\(x < -4\)[/tex]
- [tex]\(-4 < x < \frac{1}{2}\)[/tex]
- [tex]\(x > \frac{1}{2}\)[/tex]
Test the intervals:
1. Interval [tex]\(x < -4\)[/tex]: Choose a test point like [tex]\(x = -5\)[/tex].
[tex]\[
\frac{x+4}{2x-1} = \frac{-5+4}{2(-5)-1} = \frac{-1}{-11} = \frac{1}{11} > 0
\][/tex]
This interval does not satisfy the inequality.
2. Interval [tex]\(-4 < x < \frac{1}{2}\)[/tex]: Choose a test point like [tex]\(x = 0\)[/tex].
[tex]\[
\frac{x+4}{2x-1} = \frac{0+4}{2(0)-1} = \frac{4}{-1} = -4 < 0
\][/tex]
This interval satisfies the inequality.
3. Interval [tex]\(x > \frac{1}{2}\)[/tex]: Choose a test point like [tex]\(x = 1\)[/tex].
[tex]\[
\frac{x+4}{2x-1} = \frac{1+4}{2(1)-1} = \frac{5}{1} = 5 > 0
\][/tex]
This interval does not satisfy the inequality.
### Step 3: Formulate the solution.
The solution is the interval where the inequality is true. From our testing, this is:
[tex]\(-4 < x < \frac{1}{2}\)[/tex]
Thus, the solution is [tex]\(-4 < x < \frac{1}{2}\)[/tex].
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