Solving Inequalities

Solving Inequalities

I was going through the posts on inequalities and found that many good concepts are explained here, but still people do have trouble solving the question using these concept.
In these posts, there were quadratic equations, curves, graphs and other mathematical stuff. With this post, I am trying to provide a simple method to solve such questions quickly. I won't be writing the concepts behind it.
Remember this is the same OLD concept, it's just presented differently.

Case 1: Multiplication

for example: \((x-1)(x-2)(x-3)(x-7) \leq 0\)

To check the intervals in which this inequality holds true, we need to pick only one value from the number line.
Lets say x = 10, then (9)(8)(7)(3) > 0, in every alternate interval the sign would be + for the above expression

---(+)-----1---(-)---2---(+)---3-----------(-)----------7----(+)------

Thus, inequality would hold true in the intervals:
\(1 \leq x \leq 2\)
\(3 \leq x \leq 7\),

Note that intervals are inclusive of 3 & 7

Case 2: Division

In case of division:
\(\frac{(x-1)(x-2)}{(x-3)(x-7)} \leq 0\)
Using the same approach as above;
\(1 \leq x \leq 2\)
\(3 < x < 7,\) \((x\neq3,\) \(7)\)

Since (x-3)(x-7) is in denominator, its value can't be 0.


Following things to be kept in mind while using above method:
1. Cofficient of x should be positive: for ex - \((x-a)(b-x)>0\), can be written as \((x-a)(x-b)<0\)
2. Even powers: for ex - \((x-9)^2(x+3) \geq 0\), \((x-9)^2\) is always greater than 0, so, it should be only considered to check the equality (=0)
3. Odd powers: \((x-a)^3(x-b)^5>0\), will be same as \((x-a)(x-b)>0\)
4. Cancelling the common terms:

for ex - \(\frac {(x^2+x-6)(x-11)}{(x+3)} >0\), it can be simplified as (x-2)(x-11)>0
or, ---(+)-----2---(-)-------------11----(+)------
thus \(x <2\) or \(x>11\), but since at x = -3 (in the original expression), we get undefined form, so, \(x \\
\\
\neq -3\)

A question for you:
For what values of x, does the following inequality holds true?
\((x-a)(x-b)...(x-n)...(x-z) \geq 0\), where {a, b, c,...} are integers.

Reference post:
https://gmatclub.com/forum/inequalities-trick-91482.html

PS: I hope you find this post useful, please provide feedback to improve the quality of the post.

Thanks,