Solve the following inequality:

Question

2/x-2≤x-1

Please explain in a detailed step-by-step process, I am very lost. Thank You.

Answer 1

Here us a Desmos graph that confirms the solution set shown in the video.

desmos.com/calculator/ybqau0k9lz

Answer 2

It is possible to break down the problem into two cases, but my memory is bad, and I'd rather keep the number of things I have to remember to a minimum.

One thing to keep in mind, your denominator cannot be 0 so x ≠ 2.

2 / (x-2) ≤ x -1

We just move everything to the left side, so:

2 / (x-2) - (x - 1) ≤ 0

2 / (x-2) - (x - 1)(x-2) / (x-2) ≤ 0

2 / (x-2) - (x² - x - 2x + 2) / (x-2) ≤ 0

[2 - (x² - x - 2x + 2)] / (x-2) ≤ 0

(2 - x² + x + 2x - 2) / (x-2) ≤ 0

(3x-x²) / (x-2) ≤ 0

x(3-x) / (x-2) ≤ 0

At this point, I create a table to sample the function at its points of interests (zeros, asymptotes) and between. I just want to see where it is positive, zero, or negative.

(Sorry, tabs don't seem to work in this editor.)

x value | nominator | denominator | overall

-1 | - | - | +

0 | 0 | - | 0

1 | + | - | -

2 | + | asymptote | asymptote

2.5 | + | + | +

3 | 0 | + | 0

4 | - | + | -

As you can see, the function is overall ≤ 0 only at 0 ≤ x < 2 and x ≥ 3. Note that this also satisfies our earlier requirement that x ≠ 2.