How do you solve absolute value inequalities with absolute value variables on both sides #abs(2x)<=abs(x-3)#?

1 Answer
bp
Apr 7, 2015

All values of x in the interval [-3,2]

To remove the absolute value sign, square both sides, so that both would be positive only. Accordingly,
4#x^2# #<=# #x-3)^2#
4#x^2# #<=# #x^2# -6x+9
3#x^2# +6x - 9 #<=0#
#x^2# +2x -3 #<=#0
(x+3)(x-2) #<=0#.

Now, there can be two options for this inequality to hold good.
Case 1 x+3 is positive, that is x #>=#-3. and x-2 is negative, that is x#<=#2.

Case 2 x+3 is negative, that is x#<=# -3 and x-2 is positive, that is x#>=#2

To understand the solution mark the above inequalities on the number line. The solution to the inequality would be x#>=#-3 and x#<=#2. In the interval notation it would be [-3,2]

Related questions
See all questions in Absolute Value Inequalities
Impact of this question
6518 views around the world
You can reuse this answer
Creative Commons License