Solution - Absolute value equations

$$\left(-5x+1\right)+x=\left(-x+2\right)+x$$

Group like terms:

$$\left(-5x+x\right)+1=\left(-x+2\right)+x$$

Simplify the arithmetic:

$$-4x+1=\left(-x+2\right)+x$$

Group like terms:

$$-4x+1=\left(-x+x\right)+2$$

Simplify the arithmetic:

$$-4x+1=2$$

Subtract $$$$ from both sides:

$$\left(-4x+1\right)-1=2-1$$

Simplify the arithmetic:

$$-4x=2-1$$

Simplify the arithmetic:

$$-4x=1$$

Divide both sides by :

$$\frac{\left(-4x\right)}{-4}=\frac{1}{-4}$$

Cancel out the negatives:

$$\frac{4x}{4}=\frac{1}{-4}$$

Simplify the fraction:

$$x=\frac{1}{-4}$$

Move the negative sign from the denominator to the numerator:

$$x=\frac{-1}{4}$$

$$\left(-5x+1\right)=x-2$$

Subtract $$$$ from both sides:

$$\left(-5x+1\right)-x=\left(x-2\right)-x$$

Group like terms:

$$\left(-5x-x\right)+1=\left(x-2\right)-x$$

Simplify the arithmetic:

$$-6x+1=\left(x-2\right)-x$$

Group like terms:

$$-6x+1=\left(x-x\right)-2$$

Simplify the arithmetic:

$$-6x+1=-2$$

Subtract $$$$ from both sides:

$$\left(-6x+1\right)-1=-2-1$$

Simplify the arithmetic:

$$-6x=-2-1$$

Simplify the arithmetic:

$$-6x=-3$$

Divide both sides by :

$$\frac{\left(-6x\right)}{-6}=\frac{-3}{-6}$$

Cancel out the negatives:

$$\frac{6x}{6}=\frac{-3}{-6}$$

Simplify the fraction:

$$x=\frac{-3}{-6}$$

Cancel out the negatives:

$$x=\frac{3}{6}$$

Find the greatest common factor of the numerator and denominator:

$$x=\frac{\left(1·3\right)}{\left(2·3\right)}$$

Factor out and cancel the greatest common factor:

$$x=\frac{1}{2}$$