To solve the inequality [tex]\(\left|\frac{2}{x-4}\right| > 1\)[/tex], [tex]\(x \neq 4\)[/tex], we need to consider the definition of the absolute value function and the properties of inequalities. Let's proceed step by step:
1. Understanding the Absolute Value Inequality:
[tex]\[\left|\frac{2}{x-4}\right| > 1\][/tex]
This inequality means that the expression [tex]\(\frac{2}{x-4}\)[/tex] is either greater than 1 or less than -1.
2. Breaking the Inequality into Two Cases:
We need to solve the two inequalities separately:
Case 1: [tex]\[\frac{2}{x-4} > 1\][/tex]
Case 2: [tex]\[\frac{2}{x-4} < -1\][/tex]
3. Solving Case 1: [tex]\(\frac{2}{x-4} > 1\)[/tex]
- Multiply both sides by [tex]\(x-4\)[/tex]. Note that the sign of the inequality depends on whether [tex]\(x-4\)[/tex] is positive or negative.
- Case 1a: [tex]\(x-4 > 0\)[/tex] (i.e., [tex]\(x > 4\)[/tex]):
[tex]\[\frac{2}{x-4} > 1 \implies 2 > x - 4 \implies x - 4 < 2\][/tex]
[tex]\[x < 6\][/tex]
So for Case 1a, we get [tex]\(4 < x < 6\)[/tex].
4. Solving Case 2: [tex]\(\frac{2}{x-4} < -1\)[/tex]
- Multiply both sides by [tex]\(x-4\)[/tex] again, but keep in mind that the direction of the inequality flips if [tex]\(x-4\)[/tex] is positive.
- Case 2a: [tex]\(x-4 > 0\)[/tex] (i.e., [tex]\(x > 4\)[/tex]):
[tex]\[\frac{2}{x-4} < -1 \implies 2 < - (x - 4) \implies 2 < - x + 4 \implies x < 2 - 4 \implies x < 2\][/tex]
This cannot happen as it contradicts [tex]\(x > 4\)[/tex].
- Case 2b: [tex]\(x-4 < 0\)[/tex] (i.e., [tex]\(x < 4\)[/tex]):
[tex]\[\frac{2}{x-4} < -1 \implies 2 < - (x - 4) \implies 2 < - x + 4 \implies 2 < 4 - x \implies x < 2\][/tex]
5. Combining the Solutions:
From Case 1a: [tex]\(4 < x < 6\)[/tex]
From Case 2b: [tex]\(x < 2\)[/tex]
6. Exclusion of [tex]\(x = 4\)[/tex]:
- [tex]\(x \neq 4\)[/tex] is also part of the original problem, so [tex]\(x = 4\)[/tex] must be excluded from the solution set.
7. Union of the Intervals:
Combining the intervals obtained from both cases, we get:
[tex]\[(-\infty, 2) \cup (4, 6)\][/tex]
Hence, the solution to the inequality [tex]\(\left|\frac{2}{x-4}\right| > 1\)[/tex] where [tex]\(x \neq 4\)[/tex] is the set of [tex]\(x\)[/tex] values from [tex]\(x < 2\)[/tex] or [tex]\(4 < x < 6\)[/tex]. So, the final solution set with [tex]\(x \neq 4\)[/tex] is:
[tex]\[(-\infty, 2) \cup (4, 6)\][/tex]
