Question

If |x-1| + |x+ 1| = 2, find x

Please explain for the interval (-1, 1), with an explanation for each step

Answer 1

• x ∈ [ - 1, 1 ]

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There are two points that make absolute values equal to zero:

• x = - 1 and x = 1

These two points split the range into three regions:

• x < - 1, - 1 ≤ x ≤ 1 and x > 1

Let's solve the equation in each region.

1) x < - 1, both expressions inside absolute value are negative:

• - (x - 1) - (x + 1) = 2
• - x + 1 - x - 1 = 2
• -2x = 2
• x = - 1

It means no solution as - 1 is outside of the region

2) -1 ≤ x ≤ 1, one of expressions is negative and one- positive:

• - (x - 1) + (x + 1) = 2
• - x + 1 + x + 1 = 2
• 2 = 2

It means any value of x in the given interval is the solution.

3) x > 1, both expressions are positive:

• (x - 1) + (x + 1) = 2
• x - 1 + x + 1 = 2
• 2x = 2
• x = 1

No solution as 1 is outside of the region.

So the solution is x ∈ [-1, 1].