Solution - Absolute value equations

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

Break up the fraction:

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

Find the greatest common factor of the numerator and denominator:

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

Factor out and cancel the greatest common factor:

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

Multiply the fraction(s):

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

Break up the fraction:

$$\frac{x}{3}-1=\frac{x}{2}+\frac{2}{2}$$

Find the greatest common factor of the numerator and denominator:

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

Factor out and cancel the greatest common factor:

$$\frac{x}{3}-1=\frac{x}{2}+1$$

Subtract $$$$ from both sides:

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

Group like terms:

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

Group the coefficients:

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

Find the lowest common denominator:

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

Multiply the denominators:

$$\left(\frac{\left(1·2\right)}{6}+\frac{\left(-1·3\right)}{6}\right)x-1=\left(\frac{x}{2}+1\right)-\frac{x}{2}$$

Multiply the numerators:

$$\left(\frac{2}{6}+\frac{-3}{6}\right)x-1=\left(\frac{x}{2}+1\right)-\frac{x}{2}$$

Combine the fractions:

$$\frac{\left(2-3\right)}{6}x-1=\left(\frac{x}{2}+1\right)-\frac{x}{2}$$

Combine the numerators:

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

Group like terms:

$$\frac{-1}{6}·x-1=\left(\frac{x}{2}+\frac{-1}{2}x\right)+1$$

Combine the fractions:

$$\frac{-1}{6}·x-1=\frac{\left(1-1\right)}{2}x+1$$

Combine the numerators:

$$\frac{-1}{6}·x-1=\frac{0}{2}x+1$$

Reduce the zero numerator:

$$\frac{-1}{6}x-1=0x+1$$

Simplify the arithmetic:

$$\frac{-1}{6}x-1=1$$

Add $$$$ to both sides:

$$\left(\frac{-1}{6}x-1\right)+1=1+1$$

Simplify the arithmetic:

$$\frac{-1}{6}x=1+1$$

Simplify the arithmetic:

$$\frac{-1}{6}x=2$$

Multiply both sides by inverse fraction $$$$:

$$\left(\frac{-1}{6}x\right)·\frac{6}{-1}=2·\frac{6}{-1}$$

Group like terms:

$$\left(\frac{-1}{6}·-6\right)x=2·\frac{6}{-1}$$

Multiply the coefficients:

$$\frac{\left(-1·-6\right)}{6}x=2·\frac{6}{-1}$$

Simplify the arithmetic:

$$1x=2·\frac{6}{-1}$$

$$x=2·\frac{6}{-1}$$

Simplify the arithmetic:

$$x=-12$$

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

Break up the fraction:

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

Find the greatest common factor of the numerator and denominator:

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

Factor out and cancel the greatest common factor:

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

Multiply the fraction(s):

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

Expand the parentheses:

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

Break up the fraction:

$$\frac{x}{3}-1=\frac{-x}{2}+\frac{-2}{2}$$

Find the greatest common factor of the numerator and denominator:

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

Factor out and cancel the greatest common factor:

$$\frac{x}{3}-1=\frac{-x}{2}-1$$

Add $$$$ to both sides:

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

Group like terms:

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

Group the coefficients:

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

Find the lowest common denominator:

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

Multiply the denominators:

$$\left(\frac{\left(1·2\right)}{6}+\frac{\left(1·3\right)}{6}\right)x-1=\left(\frac{-x}{2}-1\right)+\frac{1}{2}x$$

Multiply the numerators:

$$\left(\frac{2}{6}+\frac{3}{6}\right)x-1=\left(\frac{-x}{2}-1\right)+\frac{1}{2}x$$

Combine the fractions:

$$\frac{\left(2+3\right)}{6}·x-1=\left(\frac{-x}{2}-1\right)+\frac{1}{2}x$$

Combine the numerators:

$$\frac{5}{6}·x-1=\left(\frac{-x}{2}-1\right)+\frac{1}{2}x$$

Group like terms:

$$\frac{5}{6}·x-1=\left(\frac{-x}{2}+\frac{1}{2}x\right)-1$$

Combine the fractions:

$$\frac{5}{6}·x-1=\frac{\left(-1+1\right)}{2}x-1$$

Combine the numerators:

$$\frac{5}{6}·x-1=\frac{0}{2}x-1$$

Reduce the zero numerator:

$$\frac{5}{6}x-1=0x-1$$

Simplify the arithmetic:

$$\frac{5}{6}x-1=-1$$

Add $$$$ to both sides:

$$\left(\frac{5}{6}x-1\right)+1=-1+1$$

Simplify the arithmetic:

$$\frac{5}{6}x=-1+1$$

Simplify the arithmetic:

$$\frac{5}{6}x=0$$

Divide both sides by the coefficient:

$$x=0$$