1 − 2|x|
1 − |x|
≤
1
2
.
Let's solve the inequality step by step:
1. First, let's consider the case when x is positive or zero.
When x ≥ 0, |x| = x.
So, the inequality becomes:
1 - 2x ≤ 1/2
Simplifying the inequality, we get:
-2x ≤ -1/2
Dividing both sides by -2 and changing the direction of the inequality:
x ≥ 1/4
Therefore, for x ≥ 0, the inequality is satisfied when x ≥ 1/4.
2. Now, let's consider the case when x is negative.
When x < 0, |x| = -x.
So, the inequality becomes:
1 - 2(-x) ≤ 1/2
Simplifying the inequality, we get:
1 + 2x ≤ 1/2
Subtracting 1 from both sides and simplifying further:
2x ≤ -1/2
Dividing both sides by 2 and changing the direction of the inequality:
x ≤ -1/4
Therefore, for x < 0, the inequality is satisfied when x ≤ -1/4.
Combining the two cases, we have:
x ≥ 1/4 or x ≤ -1/4.
So, the solution to the inequality is x ≤ -1/4 or x ≥ 1/4.