Question

solve inequality with sign chart:

-\frac{10}{x-5}\ge-\frac{11}{x-6}

Answer 1

[AI-generated answer — Bot GPT 3.5]

To solve the inequality using a sign chart, we first need to find the critical points where the inequality might change.

Setting the denominators of both fractions to zero, we have:
x - 5 = 0 -> x = 5
x - 6 = 0 -> x = 6

Now we create a sign chart with three regions based on these critical points: x < 5, 5 < x < 6, and x > 6.

In the region x < 5:
Testing x = 0:
-(-10/5) ≥ -(-11/6)
-(-2) ≥ 11/6
2 ≥ 11/6 -> 12/6 ≥ 11/6 -> 2 > 11/6, which is true.

So in this region, the inequality is true.

In the region 5 < x < 6:
Testing x = 5.5:
-(-10/0.5) ≥ -(-11/(-0.5))
-(-20) ≥ 22
20 ≥ 22, which is false.
So in this region, the inequality is false.

In the region x > 6:
Testing x = 7:
-(-10/2) ≥ -(-11/1)
-5 ≥ -(-11)
-5 ≥ 11, which is false.
So in this region, the inequality is false.

Therefore, the solution to the inequality is x < 5.