To solve the inequality [tex]\(\frac{3x + 8}{x - 4} \geq 0\)[/tex], we need to analyze the expression [tex]\(\frac{3x + 8}{x - 4}\)[/tex] and determine where it is non-negative (i.e., positive or zero).
Follow these steps:
### Step 1: Identify Critical Points
The values of [tex]\(x\)[/tex] that make the numerator or the denominator zero are critical points. These points divide the number line into different intervals that we need to test.
For the numerator [tex]\(3x + 8 = 0\)[/tex]:
[tex]\[ 3x + 8 = 0 \][/tex]
[tex]\[ 3x = -8 \][/tex]
[tex]\[ x = -\frac{8}{3} \][/tex]
For the denominator [tex]\(x - 4 = 0\)[/tex]:
[tex]\[ x - 4 = 0 \][/tex]
[tex]\[ x = 4 \][/tex]
### Step 2: Determine Sign of Each Interval
We need to test the sign of the expression [tex]\(\frac{3x + 8}{x - 4}\)[/tex] in each interval defined by the critical points [tex]\(x = -\frac{8}{3}\)[/tex] and [tex]\(x = 4\)[/tex]:
- [tex]\( (-\infty, -\frac{8}{3}) \)[/tex]
- [tex]\( (-\frac{8}{3}, 4) \)[/tex]
- [tex]\( (4, \infty) \)[/tex]
Choose a test point within each interval and substitute it into [tex]\(\frac{3x + 8}{x - 4}\)[/tex].
1. Interval [tex]\( (-\infty, -\frac{8}{3}) \)[/tex]:
- Test point [tex]\(x = -3\)[/tex]:
[tex]\[ \frac{3(-3) + 8}{-3 - 4} = \frac{-9 + 8}{-7} = \frac{-1}{-7} = \frac{1}{7} > 0 \][/tex]
- The expression is positive in this interval.
2. Interval [tex]\( (-\frac{8}{3}, 4) \)[/tex]:
- Test point [tex]\(x = 0\)[/tex]:
[tex]\[ \frac{3(0) + 8}{0 - 4} = \frac{8}{-4} = -2 < 0 \][/tex]
- The expression is negative in this interval.
3. Interval [tex]\( (4, \infty) \)[/tex]:
- Test point [tex]\(x = 5\)[/tex]:
[tex]\[ \frac{3(5) + 8}{5 - 4} = \frac{15 + 8}{1} = 23 > 0 \][/tex]
- The expression is positive in this interval.
### Step 3: Include Critical Points
- At [tex]\(x = -\frac{8}{3}\)[/tex], the numerator is zero, making the entire expression zero. Therefore, [tex]\(x = -\frac{8}{3}\)[/tex] satisfies the inequality.
- At [tex]\(x = 4\)[/tex], the denominator is zero, making the expression undefined. Therefore, [tex]\(x = 4\)[/tex] does not satisfy the inequality.
### Step 4: Combine Intervals
From the intervals and critical points analysis, the solution to the inequality [tex]\(\frac{3x + 8}{x - 4} \geq 0\)[/tex] is:
- [tex]\( x \leq -\frac{8}{3} \)[/tex]
- or [tex]\( x > 4 \)[/tex]
Thus, the solution is:
[tex]\[ x \leq -\frac{8}{3} \text{ or } x > 4 \][/tex]
