|x|=x
1) Let's solve in ℝ: |x| = x
The absolute-value arguments are zero at 0. Split ℝ at these points and solve on each interval
On (-∞, 0]: x ≤ 0, so |x| = -x
On (-∞, 0], the equation becomes -x = x
2) Let's solve in ℝ: -x = x
-x -x = 0
ⓘ Like terms are grouped by adding their coefficients: -x -x = -2x
-2x = 0
We carry out the simplification of the expression by -2
⇥We solve in ℝ: x = 0
⇥Solution obtained: x = 0
As 0 ≤ 0, we can keep the solution 0
On [0, +∞): x ≥ 0, so |x| = x
On [0, +∞), the equation becomes x = x
3) Let's solve in ℝ: x = x
x -x = 0
ⓘ Group like terms by adding their coefficients: x -x = 0
0 = 0
Solution found: x ∈ ℝ
The equation holds for every x in [0, +∞): keep the whole interval
▶Solution found: x ≥ 0
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