Let's analyze the problem step-by-step to determine the correct inequality that can be used to find the width [tex]\( w \)[/tex] of the tree house base such that the area is greater than 293 square inches.
1. Understanding the Variables:
- Let [tex]\( w \)[/tex] represent the width of the rectangular base in inches.
- According to the problem, the length of the base is 7 inches more than its width. Thus, the length can be expressed as [tex]\( w + 7 \)[/tex].
2. Expression for Area:
- The area [tex]\( A \)[/tex] of a rectangle is given by the formula:
[tex]\[
A = \text{width} \times \text{length}
\][/tex]
- Substituting the expressions for width and length:
[tex]\[
A = w \times (w + 7)
\][/tex]
3. Area Condition:
- We are given that the area of the base should be greater than 293 square inches. Therefore, we set up the inequality:
[tex]\[
w \times (w + 7) > 293
\][/tex]
4. Simplifying the Inequality:
- Expanding the expression on the left side:
[tex]\[
w(w + 7) = w^2 + 7w
\][/tex]
- Thus, the inequality becomes:
[tex]\[
w^2 + 7w > 293
\][/tex]
Therefore, the inequality that determines the width [tex]\( w \)[/tex] such that the area of the base of the tree house is greater than 293 square inches is:
[tex]\(\boxed{w^2 + 7w > 293}\)[/tex]
