How do you solve the equation $| 2 x - 1 | = x + 1$ $abs(2x-1)=x+1$ ?

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Answer 1 · 2017-08-15T11:04:54

See below.

$| 2 x - 1 | = x - \frac{1}{2} + \frac{3}{2} = \frac{1}{2} (2 x - 1) + \frac{3}{2}$ $abs(2x-1)=x-1/2+3/2 = 1/2(2x-1)+3/2$

now assuming $2 x - 1 \neq 0$ $2x-1 ne 0$ we have

$\frac{| 2 x - 1 |}{2 x - 1} = \frac{1}{2} + \frac{3}{2} \frac{1}{2 x - 1}$ $abs(2x-1)/(2x-1) = 1/2+3/2 1/(2x-1)$ so

$2 \frac{| 2 x - 1 |}{2 x - 1} = 1 + \frac{3}{2 x - 1}$ $2abs(2x-1)/(2x-1) = 1+3/(2x-1)$

but $\frac{| 2 x - 1 |}{2 x - 1} = \pm 1$ $abs(2x-1)/(2x-1) = pm 1$ so we have two options

${(-2=1+3/(2x-1)->-3=3/(2x-1)->2x-1=-1 -> x = 0),(2=1+3/(2x-1)->1=3/(2x-1)->2x-1=3->x=2):}$

so the solutions are

$x = {0, 2}$

How do you solve the equation abs(2x-1)=x+1|2x−1|=x+1abs(2x-1)=x+1?