What is the solution? |x-3|≤|x+7|

Question

I looked on mathway and it said no solution but I've seen other sources saying All Real Numbers. Is this extraneous??? Please provide a solution and explanation

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Answer 1 · 2017-12-09T23:39:37

#x ge -2#

#abs(x-3) le abs(x+7)# is equivalent to

#sqrt((x-3)^2) le sqrt((x+7)^2)# now squaring both sides

#(x-3)^2 le (x+7)^2# or

#x^2-6x+9 le x^2+14x+49# or

#0 le 20 x +40 rArr x ge -2#

Answer 2 · 2017-12-10T00:12:25

For another method, please see below.

Solve the equation: #abs(x-3) = abs(x+7)#

This involves solving
#x-3 = +-(x+7)#

#x+3=x+7# has no solutions.

#x-3 = -(x+7)# requires

#x-3 = -x-7# so

#2x=-4# and #x = -2#

Not test the inequality on each piece of the number line.

For #x < -2# (say #x=-5#), the inequality is not true:

#abs(-5-3) = 8##" "#is not less that #abs(-5+7) = 2#

For #-2 < x# (say #x=0#, the inequality is true:

#abs(0-3) = 3 <= abs(0+7) = 7#

The solution set is #-2 <= x#