Determine the intervals in which the following inequality is satisfied: $|1-x|-x\geq 0$

Question

Exercise:

Determine the intervals in which the following inequality is satisfied: $$|1-x|-x\geq 0$$

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Attempt:

What to Expect:

A quick manipulation renders the following: $|1-x|\geq x$.

Graphing both sides:

Eyeballing, the answer seems to be: $x \leq \frac{1}{2}$.

Solution:

(1) $|1-x|-x\geq 0$

(2) $|1-x| \geq x$

(3)

• when $x \geq 0$:
  • when $1-x \geq 0$: $1-x \geq x$
  • when $1-x < 0$: $-(1-x) \geq x$
• when $x < 0$: $x-1 > 0$

(4)

• when $x \geq 0$:
  • when $1 \geq x$: $\frac{1}{2} \geq x$
  • when $1 < x$: invalid
• when $x < 0$: $x > 1$

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Request:

I do see the expected answer in (4), but according to my solution it's only applicable when $x \geq 0$. When $x < 0$, I get an answer that seems to have no resemblance in the expected answer. Where and what did I do wrong?

Answer 1

When $x \ge 1$, $$|1-x|=x-1\ge x$$ and there is no solution.

When $x \le 1$, $$|1-x|=1-x\ge x$$ $$1\ge 2x$$ $$1/2\ge x$$

Answer 2

There are three cases:

$x=1$

the inequation becomes $-1\geq 0$ which is false.

$x>1$

it becomes

$x-1-x=-1\geq 0$ . it is false

$x<1$

it gives $1-x-x\geq 0$ and $x\leq \frac{1}{2}$

the inequation is satisfied in $(-\infty,\frac{1}{2}]$.

Answer 3

$$|1-x|-x\geq 0$$

$$\Leftrightarrow |1-x|\geq x $$

$$ \Leftrightarrow [(1-x)\geq x] \text{ or } [(1-x)\le -x]$$

$$ \Leftrightarrow (1\geq 2x) \text{ or } (1 \le 0)$$

$$ \Leftrightarrow \frac{1}{2}\geq x \text{ or } 1 \le 0$$

$$ \Leftrightarrow x \le \frac{1}{2} \text{ or } 1 \le 0$$

$$ \Leftrightarrow x \in ]-\infty, \frac{1}{2}] \text{ or } x \in \emptyset $$

$$ \Leftrightarrow x \in ]-\infty, \frac{1}{2}] \cup \emptyset $$

$$ \Leftrightarrow x \in ]-\infty, \frac{1}{2}]$$