\[{x}^{2}-3x+2>0\]
1
Factor x2−3x+2{x}^{2}-3x+2.
(x−2)(x−1)>0(x-2)(x-1)>0

2
Solve for xx.
x=2,1x=2,1

3
From the values of xx above, we have these 3 intervals to test.
x<11<x<2x>2\begin{aligned}&x<1\\&1<x<2\\&x>2\end{aligned}

4
Pick a test point for each interval.
For the interval x<1x<1:

Let's pick x=0x=0. Then, 02−3×0+2>0{0}^{2}-3\times 0+2>0.
After simplifying, we get 2>02>0, which is
true
.
Keep this interval.
.

For the interval 1<x<21<x<2:

Let's pick x=32x=\frac{3}{2}. Then, (32)2−3×32+2>0{(\frac{3}{2})}^{2}-3\times \frac{3}{2}+2>0.
After simplifying, we get −0.25>0-0.25>0, which is
false
.
Drop this interval.
.

For the interval x>2x>2:

Let's pick x=3x=3. Then, 32−3×3+2>0{3}^{2}-3\times 3+2>0.
After simplifying, we get 2>02>0, which is
true
.
Keep this interval.
.

5
Therefore,
x<1,x>2x<1,x>2

Done
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