(x−1)(x−4)>0(x-1)(x-4)>0
1
Solve for xx.
x=1,4x=1,4

2
From the values of xx above, we have these 3 intervals to test.
x<11<x<4x>4\begin{aligned}&x<1\\&1<x<4\\&x>4\end{aligned}

3
Pick a test point for each interval.
For the interval x<1x<1:

Let's pick x=0x=0. Then, (0−1)(0−4)>0(0-1)(0-4)>0.
After simplifying, we get 4>04>0, which is
true
.
Keep this interval.
.

For the interval 1<x<41<x<4:

Let's pick x=2x=2. Then, (2−1)(2−4)>0(2-1)(2-4)>0.
After simplifying, we get −2>0-2>0, which is
false
.
Drop this interval.
.

For the interval x>4x>4:

Let's pick x=5x=5. Then, (5−1)(5−4)>0(5-1)(5-4)>0.
After simplifying, we get 4>04>0, which is
true
.
Keep this interval.
.

4
Therefore,
x<1,x>4x<1,x>4

Done
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