Question

Solve the Inequality by using the Test Point Method
3/2x+5 - 1/x-1 ≤ 0

Answer 1

1. First acknowledge that

\(x \ne -\frac{5}{2} \) and \(x \ne 1\)

-5/2 and 1 are your first 2 test points.

Solve this inequality to get the third test point:


\(\frac{3}{2x+5} \le \frac{1}{x-1}\)

Answer 2

By cross multiplying, you get

3(x-1) ≤ 2x + 5

3x - 3 ≤ 2x + 5

x ≤ 8

Which is the third test point.

Answer 3

Place all the test points on the number line:

Answer 4

We know that x ≤ 8, but x cannot include 1, so we shade that segment. Since The shaded parts cannot overlap, we know the region between -5/2 and 1 does not include the solution. Therefore, the other shaded portion must be x < -5/2