What is the solution? |x-3|≤|x+7|

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

2 Answers
Dec 9, 2017

x ge -2

Explanation:

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

Dec 10, 2017

For another method, please see below.

Explanation:

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