Questions: Solve the inequality. Write the solution set in interval notation and graph the solution set. (x+9)/(x-1)<0 What is the solution set?

Transcript text: Solve the inequality. Write the solution set in interval notation and graph the solution set. \[ \frac{x+9}{x-1}<0 \] What is the solution set? $\square$ (Type your answer in interval notation.)

Solution

Solution Steps

Step 1: Identify the critical points

The inequality to solve is: \[ \frac{x+9}{x-1}<0 \] The critical points are where the numerator and denominator are zero:

$x + 9 = 0 \Rightarrow x = -9$
$x - 1 = 0 \Rightarrow x = 1$

Step 2: Determine the intervals

The critical points divide the number line into three intervals:

$(- \infty, -9)$
$(-9, 1)$
$(1, \infty)$

Step 3: Test the intervals

We test a point in each interval to determine where the inequality holds:

For $x \in (- \infty, -9)$, choose $x = -10$: \[ \frac{-10 + 9}{-10 - 1} = \frac{-1}{-11} = \frac{1}{11} > 0 \]
For $x \in (-9, 1)$, choose $x = 0$: \[ \frac{0 + 9}{0 - 1} = \frac{9}{-1} = -9 < 0 \]
For $x \in (1, \infty)$, choose $x = 2$: \[ \frac{2 + 9}{2 - 1} = \frac{11}{1} = 11 > 0 \]

Final Answer

The solution set is: \[ (-9, 1) \]

{"axisType": 3, "coordSystem": {"xmin": -10, "xmax": 10, "ymin": -10, "ymax": 10}, "commands": ["y = (x+9)/(x-1)"], "latex_expressions": ["$\\frac{x+9}{x-1}$"]}

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