Solve the inequality. Write the solution set in interval notation.
-$\frac { 4 } { x - 2 }

Question

Solve the inequality. Write the solution set in interval notation.
-$\frac { 4 } { x - 2 } < 1$
A) $( - \infty , 2 ] \cup [ 6 , \infty )$
B) $( - \infty , 2 ) \cup ( 6 , \infty )$
C) $( - \infty , 2 )$
D) $( 2,6 )$

Quizplus · Accepted Answer

To solve the inequality $\frac{4}{x-2} < 1$, we first find a common denominator to subtract 1 from both sides, leading to $\frac{4}{x-2} - \frac{x-2}{x-2} < 0$, which simplifies to $\frac{6-x}{x-2} < 0$. The critical points are x = 2 and x = 6, where the expression is undefined or equals zero. Testing intervals around these points, we find that the inequality holds for $x < 2$ and $x > 6$, excluding the points themselves since the original expression is undefined at x = 2 and not less than 1 at x = 6. Thus, the solution in interval notation is $(-\infty, 2) \cup (6, \infty)$.

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