Solution - Absolute value equations

$$\left(x-3\right)-3x=\left(3x+1\right)-3x$$

Group like terms:

$$\left(x-3x\right)-3=\left(3x+1\right)-3x$$

Simplify the arithmetic:

$$-2x-3=\left(3x+1\right)-3x$$

Group like terms:

$$-2x-3=\left(3x-3x\right)+1$$

Simplify the arithmetic:

$$-2x-3=1$$

Add $$$$ to both sides:

$$\left(-2x-3\right)+3=1+3$$

Simplify the arithmetic:

$$-2x=1+3$$

Simplify the arithmetic:

$$-2x=4$$

Divide both sides by :

$$\frac{\left(-2x\right)}{-2}=\frac{4}{-2}$$

Cancel out the negatives:

$$\frac{2x}{2}=\frac{4}{-2}$$

Simplify the fraction:

$$x=\frac{4}{-2}$$

Move the negative sign from the denominator to the numerator:

$$x=\frac{-4}{2}$$

Find the greatest common factor of the numerator and denominator:

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

Factor out and cancel the greatest common factor:

$$x=-2$$

$$\left(x-3\right)=-3x-1$$

Add $$$$ to both sides:

$$\left(x-3\right)+3x=\left(-3x-1\right)+3x$$

Group like terms:

$$\left(x+3x\right)-3=\left(-3x-1\right)+3x$$

Simplify the arithmetic:

$$4x-3=\left(-3x-1\right)+3x$$

Group like terms:

$$4x-3=\left(-3x+3x\right)-1$$

Simplify the arithmetic:

$$4x-3=-1$$

Add $$$$ to both sides:

$$\left(4x-3\right)+3=-1+3$$

Simplify the arithmetic:

$$4x=-1+3$$

Simplify the arithmetic:

$$4x=2$$

Divide both sides by :

$$\frac{\left(4x\right)}{4}=\frac{2}{4}$$

Simplify the fraction:

$$x=\frac{2}{4}$$

Find the greatest common factor of the numerator and denominator:

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

Factor out and cancel the greatest common factor:

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