1. ## word problem

if the length and the width of a rectangle are increased by two inches, the area of the rectangle is 120 square inches. if the length and the width of a rectangle are decreased by two inches, the area of the rectangle is 48 square inches. find the length of the rectangle.

2. Hello, mojojojo!

If the length and the width of a rectangle are increased by two inches,
the area of the rectangle is 120 square inches.
If the length and the width of a rectangle are decreased by two inches,
the area of the rectangle is 48 square inches.
Find the length of the rectangle.

Let $\displaystyle L$ = length of the rectangle.
Let $\displaystyle W$ = width of the rectangle.

Then we have: .$\displaystyle \begin{array}{ccccc}(L+2)(W+2) &=& 120 & [1] \\ (L-2)(W-2) &=& 48 & [2]\end{array}$

$\displaystyle \begin{array}{cccccc} \text{From [1] we have:} & LW + 2L + 2W + 4 &=& 120 \\ \text{From [2] we have:} & LW - 2L - 2W + 4 &=& 48 \end{array}$

. . Subtract: . $\displaystyle 4L + 4W \:=\:72 \quad\Rightarrow\quad W \:=\:18-L$

Substitute into [1]: . $\displaystyle (L+2)(20-L) \:=\:120 \quad\Rightarrow\quad L^2 - 18L + 80 \:=\:0$

. . $\displaystyle (L-10)(L-8) \:=\:0 \quad\Rightarrow\quad L \;=\;10,\:8$

Assuming $\displaystyle L > W$ we have: .$\displaystyle L \:=\:10\text{ inches.}$