QuestionSolve. $\frac{x+4}{x-2}<1$
The solution set is $\square$ (Type your answer in interval notation.)

Studdy Solution

STEP 1

1. The inequality $\frac{x+4}{x-2}<1$ is a rational inequality.
2. We need to find the values of $x$ that satisfy the inequality.
3. The expression is undefined when the denominator is zero.

STEP 2

1. Identify the values that make the denominator zero.
2. Rearrange the inequality to compare with zero.
3. Determine the critical points.
4. Test intervals to find the solution set.

STEP 3

Identify the values that make the denominator zero, as these are points where the expression is undefined:
$x - 2 = 0$ $x = 2$

STEP 4

Rearrange the inequality to compare with zero by subtracting 1 from both sides:
$\frac{x+4}{x-2} < 1$ $\frac{x+4}{x-2} - 1 < 0$
Combine the terms on the left-hand side over a common denominator:
$\frac{x+4}{x-2} - \frac{x-2}{x-2} < 0$ $\frac{x+4 - (x-2)}{x-2} < 0$
Simplify the numerator:
$\frac{x+4 - x + 2}{x-2} < 0$ $\frac{6}{x-2} < 0$

STEP 5

Determine the critical points by setting the numerator and denominator equal to zero:
Numerator: $6 = 0$ (This does not provide any critical points.)
Denominator: $x - 2 = 0$ $x = 2$
The critical point is $x = 2$ .

STEP 6

Test intervals defined by the critical point $x = 2$ to determine where the inequality holds:
1. Interval: $(-\infty, 2)$ - Choose a test point, e.g., $x = 0$ : $\frac{6}{0-2} = \frac{6}{-2} = -3 < 0$ - The inequality holds.
2. Interval: $(2, \infty)$ - Choose a test point, e.g., $x = 3$ : $\frac{6}{3-2} = \frac{6}{1} = 6 > 0$ - The inequality does not hold.
The solution set is the interval where the inequality holds:
$(-\infty, 2)$
The solution set is:
$\boxed{(-\infty, 2)}$

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