Quantitative Reasoning

One often encounters expressions such as:
(x - a)(x - b)(x - c)...(x -k) > 0 or < 0
in which we have to solve for x.

Critical Points Method is the easiest way to solve such questions once you learn it.
Let me teach you the method here.

for the function f(x) = (x - a)(x - b)(x - c)...(x - k), Critical Points are those points at which f(x) = 0. So a, b, c.. k are the critical points of f(x).

Example 1:
Solve
(x + 1).(x - 2) < 0
Step 1. Find out the critical points
The critical points are -1 and 2 in this case.
Step 2. Plot them on a number line in a proper order.



Step 3: Consider the region above the number line to be positive and below it to be negative. Start drawing a curve starting from the right most - top side as shown below



And then complete the curve in an alternate up-down fashion as shown below




Step 4: Mark '+' in the region above the number line and '-' in the region below the number line as shown below.



In the '+' region f(x) > 0 and in the '-' region f(x) < 0.

Since the question wants us to solve for f(x) < 0, the values of x for which the curve goes below the number line will be chosen.

The shaded region shown below is our answer.



Answer: -1 < x < 2

Example 2: Lets take another question.
(x - 3)(x - 2)(x - 1) > 0

Step 1: The critical points are 1, 2 and 3.
Step 2: Mark them on a Number Line.
Step 3: Plot the curve starting from top-right side in an alternate fashion.
Step 4: Mark alternate '+' and '-' and choose the appropriate region
In this case the region in which f(x) > 0 is favourable to us is shown below



Answer: 1 < x < 2 , x > 3

Note:
(i) If the expression is in the for (a - x)(x - b)(x - c) form, first convert it into (x - a)(x - b)(x - c) form by multiplying both sides of the inequality by -1 (and flipping the inequality sign). Apply the Critical Points method only after that.

For example:
(1 - 2x)(1 + x) < 0
has to be first converted into
(2x - 1)(x + 1) > 0 form.

(ii) If the inequality contains an 'equal to' sign as well such as (x - a)(x - b)(x - c) >=0 or =<0,
we have to include the critical points in our region as a part of our solution.