Let's solve the rational inequality [tex]\(\frac{-2}{1-x} < 8\)[/tex] step-by-step.
1. Rewrite the Inequality:
We start with:
[tex]\[
\frac{-2}{1-x} < 8
\][/tex]
2. Multiply Both Sides by the Denominator:
The inequality involves the fraction [tex]\(\frac{-2}{1-x}\)[/tex]. To eliminate the fraction, we multiply both sides of the inequality by [tex]\(1 - x\)[/tex]. Note that we must consider the sign of [tex]\(1 - x\)[/tex] because it affects the direction of the inequality.
- When [tex]\(1 - x > 0\)[/tex] (or [tex]\(x < 1\)[/tex]), we multiply both sides by [tex]\(1 - x\)[/tex], and the direction of the inequality remains the same:
[tex]\[
-2 < 8(1 - x)
\][/tex]
- When [tex]\(1 - x < 0\)[/tex] (or [tex]\(x > 1\)[/tex]), we multiply both sides by [tex]\(1 - x\)[/tex], and the direction of the inequality reverses:
[tex]\[
-2 > 8(1 - x)
\][/tex]
3. Consider Each Case Separately:
- Case 1: [tex]\(1 - x > 0\)[/tex] or [tex]\(x < 1\)[/tex]
[tex]\[
-2 < 8(1 - x)
\][/tex]
Simplify the right side:
[tex]\[
-2 < 8 - 8x
\][/tex]
Subtract 8 from both sides:
[tex]\[
-2 - 8 < -8x
\][/tex]
Simplify further:
[tex]\[
-10 < -8x
\][/tex]
Divide by -8 (remember to reverse the inequality sign):
[tex]\[
\frac{-10}{-8} > x \quad \text{or} \quad \frac{5}{4} > x
\][/tex]
Therefore:
[tex]\[
x < \frac{5}{4}
\][/tex]
- Case 2: [tex]\(1 - x < 0\)[/tex] or [tex]\(x > 1\)[/tex]
[tex]\[
-2 > 8(1 - x)
\][/tex]
Simplify the right side:
[tex]\[
-2 > 8 - 8x
\][/tex]
Subtract 8 from both sides:
[tex]\[
-2 - 8 > -8x
\][/tex]
Simplify further:
[tex]\[
-10 > -8x
\][/tex]
Divide by -8 (remember to reverse the inequality sign):
[tex]\[
\frac{-10}{-8} < x \quad \text{or} \quad \frac{5}{4} < x
\][/tex]
Therefore:
[tex]\[
x > \frac{5}{4}
\][/tex]
4. Combine the Results:
We have two cases:
[tex]\[
x < \frac{5}{4} \quad \text{and} \quad x > \frac{5}{4}
\][/tex]
However, since [tex]\(x\)[/tex] cannot equal [tex]\(\frac{5}{4}\)[/tex] (it would make the denominator zero), we exclude [tex]\(\frac{5}{4}\)[/tex].
Therefore, combining the results gives us all real numbers but excluding [tex]\(x = \frac{5}{4}\)[/tex]:
[tex]\[
x \in (-\infty, \frac{5}{4}) \cup (\frac{5}{4}, \infty)
\][/tex]
The Solution Set in Interval Notation:
[tex]\[
(-\infty, \frac{5}{4}) \cup (\frac{5}{4}, \infty)
\][/tex]
