[45] The solution set of the equation ∣𝑥–1∣=𝑥–1∣x–1∣=x–1 is the set of all values of 𝑥x such that:
Question
The solution set of the equation
|x - 1| = x - 1
|x - 1| = x - 1
is the set of all values of x such that:
Solution
The equation given is |x - 1| = x - 1.
The absolute value function |x - 1| can be split into two cases:
-
When (x - 1) is positive or zero, |x - 1| = x - 1. This means x - 1 ≥ 0, so x ≥ 1.
-
When (x - 1) is negative, |x - 1| = -(x - 1). However, this case will not give us a solution because if we substitute |x - 1| with -(x - 1) in the original equation, we get -(x - 1) = x - 1, which simplifies to -x + 1 = x - 1. This further simplifies to 2x = 2, which gives x = 1. But this contradicts our assumption that x < 1 for this case.
So, the solution set of the equation |x - 1| = x - 1 is the set of all x such that x ≥ 1.
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