Solution - Absolute value equations

$$\left(z+\frac{3}{8}\right)-z=\left(z+\frac{-7}{8}\right)-z$$

Group like terms:

$$\left(z-z\right)+\frac{3}{8}=\left(z+\frac{-7}{8}\right)-z$$

Simplify the arithmetic:

$$\frac{3}{8}=\left(z+\frac{-7}{8}\right)-z$$

Group like terms:

$$\frac{3}{8}=\left(z-z\right)+\frac{-7}{8}$$

Simplify the arithmetic:

$$\frac{3}{8}=\frac{-7}{8}$$

The statement is false:

$$\frac{3}{8}=\frac{-7}{8}$$

$$\left(z+\frac{3}{8}\right)=-z+\frac{7}{8}$$

Add $$$$ to both sides:

$$\left(z+\frac{3}{8}\right)+z=\left(-z+\frac{7}{8}\right)+z$$

Group like terms:

$$\left(z+z\right)+\frac{3}{8}=\left(-z+\frac{7}{8}\right)+z$$

Simplify the arithmetic:

$$2z+\frac{3}{8}=\left(-z+\frac{7}{8}\right)+z$$

Group like terms:

$$2z+\frac{3}{8}=\left(-z+z\right)+\frac{7}{8}$$

Simplify the arithmetic:

$$2z+\frac{3}{8}=\frac{7}{8}$$

Subtract $$$$ from both sides:

$$\left(2z+\frac{3}{8}\right)-\frac{3}{8}=\left(\frac{7}{8}\right)-\frac{3}{8}$$

Combine the fractions:

$$2z+\frac{\left(3-3\right)}{8}=\left(\frac{7}{8}\right)-\frac{3}{8}$$

Combine the numerators:

$$2z+\frac{0}{8}=\left(\frac{7}{8}\right)-\frac{3}{8}$$

Reduce the zero numerator:

$$2z+0=\left(\frac{7}{8}\right)-\frac{3}{8}$$

Simplify the arithmetic:

$$2z=\left(\frac{7}{8}\right)-\frac{3}{8}$$

Combine the fractions:

$$2z=\frac{\left(7-3\right)}{8}$$

Combine the numerators:

$$2z=\frac{4}{8}$$

Find the greatest common factor of the numerator and denominator:

$$2z=\frac{\left(1·4\right)}{\left(2·4\right)}$$

Factor out and cancel the greatest common factor:

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

Divide both sides by :

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

Simplify the fraction:

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

Simplify the arithmetic:

$$z=\frac{1}{\left(2·2\right)}$$

$$z=\frac{1}{4}$$