Inequalities algebraically (precalc) - help explain this to me please?

Question

Alright so the problem is as follows...

x+2 / 2x-3 > ((with line under it)) 1

I can get pretty far into solving it...

(x+2 / 2x-3) - (2x+3 / 2x +3) >line 0

5 - x / 2x - 3 >line 0

then theres where i get stuck... can someone explain where the answer

3/2 < X < 5 comes out of that?

Answer 1

„5 - x / 2x - 3 >line 0“ ... that’s wrong
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(x+2) / (2x-3) ≥ 1

First part of the solution:
Let us suppose that
2x-3 > 0
e.e.
x > 3/2.
Multiply both sides of the inequality by (2x-3):
x+2 ≥ 2x-3
-x ≥ -5
x ≤ 5
First part of the solution is
3/2 < x ≤ 5

Second part of the solution:
Let us suppose that
2x-3 < 0
e.e.
x < 3/2.
Multiply both sides of the inequality by (2x-3):
x+2 ≤ 2x-3
-x ≤ -5
x ≥ 5.
There is no cotribution to the solution.

Therefore the first part of the solution is the only solution.

Answer:
3/2 < x ≤ 5
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Note:
If there is a problem with writing "≥"
then it is better to write ">="
than "> ((with line under it))".
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