Question 1200141
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I'll do the first problem to get you started.


Let's solve the associated equation
x^2 - 5x + 4 = 0


So,
x^2 - 5x + 4 = 0
(x-1)(x-4) = 0
x-1 = 0 or x-4 = 0
x = 1 or x = 4
The quadratic formula is an alternative pathway.


The roots of x^2 - 5x + 4 are x = 1 and x = 4


Draw out a number line to plot 1 and 4 on it.


Label the following regions
A: stuff to the left of 1
B: stuff between 1 and 4
C: stuff to the right of 4


Pick a value from region A.
I'll select x = 0
Plug it into x^2 - 5x + 4 and simplify to get...
x^2 - 5x + 4 
0^2 - 5*0 + 4 
4
We arrive at a positive value.
Therefore, x = 0 makes x^2 - 5x + 4 > 0 to be true
Any other value in region A will arrive at this same conclusion.
Region A is part of the solution set.


Now onto region B.
Let's pick x = 2 as a representative value from this region 1 < x < 4
x^2 - 5x + 4
2^2 - 5*2 + 4
4 - 10 + 4
-2
The result is negative, which means x^2 - 5x + 4 > 0 would be false for any x value in region B.
Region B is NOT part of the solution set.


Lastly region C, which is x > 4
Let's pick x = 5
x^2 - 5x + 4
5^2 - 5*5 + 4
25 - 25 + 4
4
Like with region A, we get a positive result to show x^2 - 5x + 4 > 0 would be true for x values in this interval.
Region C is part of the solution set.


We found that
Region A, x < 1, is part of the solution set
Region C, x > 4, is also part of the solution set


Collectively the entire solution set is x < 1 or x > 4
We can write this in interval notation to get (-infinity, 1) U (4 < infinity)
The U symbol is the union operator. It glues together the two intervals in an "or" fashion.
We are either in the interval (-infinity, 1) OR we are in the interval (4 < infinity). We cannot be in both intervals at the same time.



Answer as inequalities: <font color=red>x < 1 or x > 4</font>
Answer as interval notation: <font color=red>(-infinity, 1) U (4 < infinity)</font>


Similar question is found here
<a href = "https://www.algebra.com/algebra/homework/Inequalities/Inequalities.faq.question.1200712.html">https://www.algebra.com/algebra/homework/Inequalities/Inequalities.faq.question.1200712.html</a>
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