Solution - Absolute value equations

$$\left(z+5\right)-z=\left(z+9\right)-z$$

Group like terms:

$$\left(z-z\right)+5=\left(z+9\right)-z$$

Simplify the arithmetic:

$$5=\left(z+9\right)-z$$

Group like terms:

$$5=\left(z-z\right)+9$$

Simplify the arithmetic:

$$5=9$$

The statement is false:

$$5=9$$

$$\left(z+5\right)=-z-9$$

Add $$$$ to both sides:

$$\left(z+5\right)+z=\left(-z-9\right)+z$$

Group like terms:

$$\left(z+z\right)+5=\left(-z-9\right)+z$$

Simplify the arithmetic:

$$2z+5=\left(-z-9\right)+z$$

Group like terms:

$$2z+5=\left(-z+z\right)-9$$

Simplify the arithmetic:

$$2z+5=-9$$

Subtract $$$$ from both sides:

$$\left(2z+5\right)-5=-9-5$$

Simplify the arithmetic:

$$2z=-9-5$$

Simplify the arithmetic:

$$2z=-14$$

Divide both sides by :

$$\frac{\left(2z\right)}{2}=\frac{-14}{2}$$

Simplify the fraction:

$$z=\frac{-14}{2}$$

Find the greatest common factor of the numerator and denominator:

$$z=\frac{\left(-7·2\right)}{\left(1·2\right)}$$

Factor out and cancel the greatest common factor:

$$z=-7$$