Solution - Absolute value equations

$$\left(\frac{3}{5}y-4\right)-\frac{2}{3}·y=\left(\frac{2}{3}y+2\right)-\frac{2}{3}y$$

Group like terms:

$$\left(\frac{3}{5}·y+\frac{-2}{3}·y\right)-4=\left(\frac{2}{3}·y+2\right)-\frac{2}{3}y$$

Group the coefficients:

$$\left(\frac{3}{5}+\frac{-2}{3}\right)y-4=\left(\frac{2}{3}·y+2\right)-\frac{2}{3}y$$

Find the lowest common denominator:

$$\left(\frac{\left(3·3\right)}{\left(5·3\right)}+\frac{\left(-2·5\right)}{\left(3·5\right)}\right)y-4=\left(\frac{2}{3}·y+2\right)-\frac{2}{3}y$$

Multiply the denominators:

$$\left(\frac{\left(3·3\right)}{15}+\frac{\left(-2·5\right)}{15}\right)y-4=\left(\frac{2}{3}·y+2\right)-\frac{2}{3}y$$

Multiply the numerators:

$$\left(\frac{9}{15}+\frac{-10}{15}\right)y-4=\left(\frac{2}{3}·y+2\right)-\frac{2}{3}y$$

Combine the fractions:

$$\frac{\left(9-10\right)}{15}·y-4=\left(\frac{2}{3}·y+2\right)-\frac{2}{3}y$$

Combine the numerators:

$$\frac{-1}{15}·y-4=\left(\frac{2}{3}·y+2\right)-\frac{2}{3}y$$

Group like terms:

$$\frac{-1}{15}·y-4=\left(\frac{2}{3}·y+\frac{-2}{3}y\right)+2$$

Combine the fractions:

$$\frac{-1}{15}·y-4=\frac{\left(2-2\right)}{3}y+2$$

Combine the numerators:

$$\frac{-1}{15}·y-4=\frac{0}{3}y+2$$

Reduce the zero numerator:

$$\frac{-1}{15}y-4=0y+2$$

Simplify the arithmetic:

$$\frac{-1}{15}y-4=2$$

Add $$$$ to both sides:

$$\left(\frac{-1}{15}y-4\right)+4=2+4$$

Simplify the arithmetic:

$$\frac{-1}{15}y=2+4$$

Simplify the arithmetic:

$$\frac{-1}{15}y=6$$

Multiply both sides by inverse fraction $$$$:

$$\left(\frac{-1}{15}y\right)·\frac{15}{-1}=6·\frac{15}{-1}$$

Group like terms:

$$\left(\frac{-1}{15}·-15\right)y=6·\frac{15}{-1}$$

Multiply the coefficients:

$$\frac{\left(-1·-15\right)}{15}y=6·\frac{15}{-1}$$

Simplify the arithmetic:

$$1y=6·\frac{15}{-1}$$

$$y=6·\frac{15}{-1}$$

Simplify the arithmetic:

$$y=-90$$

$$\left(\frac{3}{5}·y-4\right)=\frac{-2}{3}y-2$$

Add $$$$ to both sides:

$$\left(\frac{3}{5}y-4\right)+\frac{2}{3}·y=\left(\frac{-2}{3}y-2\right)+\frac{2}{3}y$$

Group like terms:

$$\left(\frac{3}{5}·y+\frac{2}{3}·y\right)-4=\left(\frac{-2}{3}·y-2\right)+\frac{2}{3}y$$

Group the coefficients:

$$\left(\frac{3}{5}+\frac{2}{3}\right)y-4=\left(\frac{-2}{3}·y-2\right)+\frac{2}{3}y$$

Find the lowest common denominator:

$$\left(\frac{\left(3·3\right)}{\left(5·3\right)}+\frac{\left(2·5\right)}{\left(3·5\right)}\right)y-4=\left(\frac{-2}{3}·y-2\right)+\frac{2}{3}y$$

Multiply the denominators:

$$\left(\frac{\left(3·3\right)}{15}+\frac{\left(2·5\right)}{15}\right)y-4=\left(\frac{-2}{3}·y-2\right)+\frac{2}{3}y$$

Multiply the numerators:

$$\left(\frac{9}{15}+\frac{10}{15}\right)y-4=\left(\frac{-2}{3}·y-2\right)+\frac{2}{3}y$$

Combine the fractions:

$$\frac{\left(9+10\right)}{15}·y-4=\left(\frac{-2}{3}·y-2\right)+\frac{2}{3}y$$

Combine the numerators:

$$\frac{19}{15}·y-4=\left(\frac{-2}{3}·y-2\right)+\frac{2}{3}y$$

Group like terms:

$$\frac{19}{15}·y-4=\left(\frac{-2}{3}·y+\frac{2}{3}y\right)-2$$

Combine the fractions:

$$\frac{19}{15}·y-4=\frac{\left(-2+2\right)}{3}y-2$$

Combine the numerators:

$$\frac{19}{15}·y-4=\frac{0}{3}y-2$$

Reduce the zero numerator:

$$\frac{19}{15}y-4=0y-2$$

Simplify the arithmetic:

$$\frac{19}{15}y-4=-2$$

Add $$$$ to both sides:

$$\left(\frac{19}{15}y-4\right)+4=-2+4$$

Simplify the arithmetic:

$$\frac{19}{15}y=-2+4$$

Simplify the arithmetic:

$$\frac{19}{15}y=2$$

Multiply both sides by inverse fraction $$$$:

$$\left(\frac{19}{15}y\right)·\frac{15}{19}=2·\frac{15}{19}$$

Group like terms:

$$\left(\frac{19}{15}·\frac{15}{19}\right)y=2·\frac{15}{19}$$

Multiply the coefficients:

$$\frac{\left(19·15\right)}{\left(15·19\right)}y=2·\frac{15}{19}$$

Simplify the fraction:

$$y=2·\frac{15}{19}$$

Multiply the fraction(s):

$$y=\frac{\left(2·15\right)}{19}$$

Simplify the arithmetic:

$$y=\frac{30}{19}$$