Question

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Which can you use to find the solution set for 2|1-x|+1≥3?
A) 1-x≥-1 AND 1-x≤1
B)1-x≤-1 OR 1-x≥1
C) 1-x≤-1 AND 1-x≥1
D) 1-x≥-1 OR 1-x≤1

Answer 1

Answer:

B)1-x≤-1 OR 1-x≥1

$S=\{x\in\mathbb{R}| x\le 0\quad \text{or}\quad x\ge 2 \}$

Step-by-step explanation:

$2|1-x|+1\geq 3$

$2|1-x|\geq 2$

$|1-x|\geq 1$

Once we have an absolute value inequality, remember that:

For a > 0

$\boxed{\text{If } |x|\leq a \Longleftrightarrow -a\leq x \leq a}$

$\boxed{\text{If } |x|\geq a \Longleftrightarrow x\leq -a \text{ or } x\geq a }$

Therefore,

$1-x\le -1\quad \text{or}\quad 1-x\ge 1$

Solving

$1-x\le -1 \Longleftrightarrow x\geq 2$

$1-x\ge 1 \Longleftrightarrow x\leq 0$

We have

$x\geq 2 \quad \cup \quad x \leq 0$

$(-\infty, 0] \cup [2, \infty)$

$S=\{x\in\mathbb{R}| x\le 0\quad \text{or}\quad x\ge 2 \}$