Writing inequality equations from word problems

The Home Cleaning Company charges $312 to power-wash the siding of a house plus $12 for

each window. Power Clean charges $36 per window, and the price includes power-washing the

siding. How many windows must a house have to make the total cost from The Home Cleaning

Company less expensive than Power Clean?

windows = x

$\begin{array}{lrcll} \text{Home Cleaning Company prize}: & 312 + 12\cdot x \\ \text{ Power Clean prize}: & 36\cdot x \\ \hline & 312 + 12\cdot x &<& \quad 36\cdot x \qquad &| \qquad - 12\cdot x\\ & 312 &<& \quad 36\cdot x - 12\cdot x\\ & 312 &<& \quad 24\cdot x\qquad &| \qquad : 24\\ & \frac{312}{24} &<& x \\ & 13 &<& x \\ & x &>& 13 \\ \end{array}$

A house must have 14 or more windows to make the total cost from The Home Cleaning

Company less expensive than Power Clean.

13 Windows:

$\begin{array}{rcl} $312 + 13\cdot $12 &=& $468 \qquad & ( \text{ Home Cleaning Company } )\\ 13\cdot $36 &=& $468 \qquad & ( \text{ Power Clean } ) \end{array}$

14 Windows:

$\begin{array}{rcl} $312 + 14\cdot $12 &=& $480 \qquad & ( \text{ Home Cleaning Company } )\\ 14\cdot $36 &=& $504 \qquad & ( \text{ Power Clean } ) \end{array}$