Question

solve inequality with sign chart:

-\frac{10}{x-5}+\frac{11}{x-6}\ge0

Answer 1

[AI-generated answer — Bot GPT 3.5]

To solve the inequality, first identify the critical points by setting the denominators equal to zero and solving for x:

x-5=0 ---> x=5
x-6=0 ---> x=6

Next, create a sign chart by plotting the critical points on a number line:

<pre>
-------------------o--o-----------
5 6
</pre>

Now, choose a test value from each interval and evaluate the inequality:

For x<5, choose x=0:
-(10/(0-5)) + (11/(0-6))
-(10/(-5)) + (11/(-6))
2 - (11/6)
2 - 1.8333 = 0.1667

For 5<x<6, choose x=5.5:
-(10/(5.5-5)) + (11/(5.5-6))
-(10/(0.5)) + (11/(-0.5))
-20 + (-22) = -42

For x>6, choose x=7:
-(10/(7-5)) + (11/(7-6))
-(10/2) + (11/1)
-5.5 + 11 = 5.5

Now, analyze the sign chart to determine where the inequality is true.

In the interval (−∞,5), the test value is positive, so the inequality is false.

In the interval (5,6), the test value is negative, so the inequality is true.

In the interval (6,∞), the test value is positive, so the inequality is true.

Therefore, the solution to the inequality is (5,6].