Solution - Absolute value equations

$$\left(\frac{7}{5}k+4\right)-\frac{1}{2}·k=\left(\frac{1}{2}k-2\right)-\frac{1}{2}k$$

Group like terms:

$$\left(\frac{7}{5}·k+\frac{-1}{2}·k\right)+4=\left(\frac{1}{2}·k-2\right)-\frac{1}{2}k$$

Group the coefficients:

$$\left(\frac{7}{5}+\frac{-1}{2}\right)k+4=\left(\frac{1}{2}·k-2\right)-\frac{1}{2}k$$

Find the lowest common denominator:

$$\left(\frac{\left(7·2\right)}{\left(5·2\right)}+\frac{\left(-1·5\right)}{\left(2·5\right)}\right)k+4=\left(\frac{1}{2}·k-2\right)-\frac{1}{2}k$$

Multiply the denominators:

$$\left(\frac{\left(7·2\right)}{10}+\frac{\left(-1·5\right)}{10}\right)k+4=\left(\frac{1}{2}·k-2\right)-\frac{1}{2}k$$

Multiply the numerators:

$$\left(\frac{14}{10}+\frac{-5}{10}\right)k+4=\left(\frac{1}{2}·k-2\right)-\frac{1}{2}k$$

Combine the fractions:

$$\frac{\left(14-5\right)}{10}·k+4=\left(\frac{1}{2}·k-2\right)-\frac{1}{2}k$$

Combine the numerators:

$$\frac{9}{10}·k+4=\left(\frac{1}{2}·k-2\right)-\frac{1}{2}k$$

Group like terms:

$$\frac{9}{10}·k+4=\left(\frac{1}{2}·k+\frac{-1}{2}k\right)-2$$

Combine the fractions:

$$\frac{9}{10}·k+4=\frac{\left(1-1\right)}{2}k-2$$

Combine the numerators:

$$\frac{9}{10}·k+4=\frac{0}{2}k-2$$

Reduce the zero numerator:

$$\frac{9}{10}k+4=0k-2$$

Simplify the arithmetic:

$$\frac{9}{10}k+4=-2$$

Subtract $$$$ from both sides:

$$\left(\frac{9}{10}k+4\right)-4=-2-4$$

Simplify the arithmetic:

$$\frac{9}{10}k=-2-4$$

Simplify the arithmetic:

$$\frac{9}{10}k=-6$$

Multiply both sides by inverse fraction $$$$:

$$\left(\frac{9}{10}k\right)·\frac{10}{9}=-6·\frac{10}{9}$$

Group like terms:

$$\left(\frac{9}{10}·\frac{10}{9}\right)k=-6·\frac{10}{9}$$

Multiply the coefficients:

$$\frac{\left(9·10\right)}{\left(10·9\right)}k=-6·\frac{10}{9}$$

Simplify the fraction:

$$k=-6·\frac{10}{9}$$

Multiply the fraction(s):

$$k=\frac{\left(-6·10\right)}{9}$$

Simplify the arithmetic:

$$k=\frac{-20}{3}$$

$$\left(\frac{7}{5}·k+4\right)=\frac{-1}{2}k+2$$

Add $$$$ to both sides:

$$\left(\frac{7}{5}k+4\right)+\frac{1}{2}·k=\left(\frac{-1}{2}k+2\right)+\frac{1}{2}k$$

Group like terms:

$$\left(\frac{7}{5}·k+\frac{1}{2}·k\right)+4=\left(\frac{-1}{2}·k+2\right)+\frac{1}{2}k$$

Group the coefficients:

$$\left(\frac{7}{5}+\frac{1}{2}\right)k+4=\left(\frac{-1}{2}·k+2\right)+\frac{1}{2}k$$

Find the lowest common denominator:

$$\left(\frac{\left(7·2\right)}{\left(5·2\right)}+\frac{\left(1·5\right)}{\left(2·5\right)}\right)k+4=\left(\frac{-1}{2}·k+2\right)+\frac{1}{2}k$$

Multiply the denominators:

$$\left(\frac{\left(7·2\right)}{10}+\frac{\left(1·5\right)}{10}\right)k+4=\left(\frac{-1}{2}·k+2\right)+\frac{1}{2}k$$

Multiply the numerators:

$$\left(\frac{14}{10}+\frac{5}{10}\right)k+4=\left(\frac{-1}{2}·k+2\right)+\frac{1}{2}k$$

Combine the fractions:

$$\frac{\left(14+5\right)}{10}·k+4=\left(\frac{-1}{2}·k+2\right)+\frac{1}{2}k$$

Combine the numerators:

$$\frac{19}{10}·k+4=\left(\frac{-1}{2}·k+2\right)+\frac{1}{2}k$$

Group like terms:

$$\frac{19}{10}·k+4=\left(\frac{-1}{2}·k+\frac{1}{2}k\right)+2$$

Combine the fractions:

$$\frac{19}{10}·k+4=\frac{\left(-1+1\right)}{2}k+2$$

Combine the numerators:

$$\frac{19}{10}·k+4=\frac{0}{2}k+2$$

Reduce the zero numerator:

$$\frac{19}{10}k+4=0k+2$$

Simplify the arithmetic:

$$\frac{19}{10}k+4=2$$

Subtract $$$$ from both sides:

$$\left(\frac{19}{10}k+4\right)-4=2-4$$

Simplify the arithmetic:

$$\frac{19}{10}k=2-4$$

Simplify the arithmetic:

$$\frac{19}{10}k=-2$$

Multiply both sides by inverse fraction $$$$:

$$\left(\frac{19}{10}k\right)·\frac{10}{19}=-2·\frac{10}{19}$$

Group like terms:

$$\left(\frac{19}{10}·\frac{10}{19}\right)k=-2·\frac{10}{19}$$

Multiply the coefficients:

$$\frac{\left(19·10\right)}{\left(10·19\right)}k=-2·\frac{10}{19}$$

Simplify the fraction:

$$k=-2·\frac{10}{19}$$

Multiply the fraction(s):

$$k=\frac{\left(-2·10\right)}{19}$$

Simplify the arithmetic:

$$k=\frac{-20}{19}$$