Solve-for-x-2-1-x-4-

Question Number 114768 by ZiYangLee last updated on 21/Sep/20

Solve for ∣ x - 2 ∣ + ∣ 1 - x ∣= 4

Answered by bobhans last updated on 21/Sep/20

(1) x < 1 \Rightarrow 2 - x + 1 - x = 4

3 - 2 x = 4; x = - \frac{1}{2}

(2) 1 ⩽ x < 2 \Rightarrow 2 - x + x - 1 = 4

1 = 4 \leftarrow n o s o l u t i o n

(3) x ⩾ 2 \Rightarrow x - 2 + x - 1 = 4

2 x = 7; x = \frac{7}{2}

∴ x = - \frac{1}{2} o r x = \frac{7}{2}

Answered by mathmax by abdo last updated on 22/Sep/20

\Rightarrow f (x) =∣ x - 2 ∣ + ∣ 1 - x ∣ - 4 = 0

x - \infty 1 2 + \infty

∣ x - 2 ∣ - x + 2 - x + 2 x - 2

∣ x - 1 ∣ - x + 1 x - 1 x - 1

f (x) - 2 x - 1 - 3 2 x - 7

\Rightarrow f (x) = {\begin{cases} - 2 x - 1 if x ⩽ 1 \\ - 3 if 1 ⩽ x ⩽ 2 \end{cases}

{2 x - 7 if x ⩾ 2

so x ⩽ 1 we get f (x) = 0 \Rightarrow - 2 x - 1 = 0 \Rightarrow x = - \frac{1}{2}

1 ⩽ x ⩽ 2 f (x) = 0 \Rightarrow - 3 = 0 imposdible

x ⩾ 2 f (x) = 0 \Rightarrow 2 x - 7 = 0 \Rightarrow x = \frac{7}{2}