Let's break down the problem step by step:
1. Define the dimensions of the wall and the paintings:
- Let [tex]\( x \)[/tex] be the width of the wall.
- The length of the wall is then [tex]\( x + 15 \)[/tex] feet.
- Each painting has dimensions of 3 feet by 4 feet, so each painting has an area of [tex]\( 3 \times 4 = 12 \)[/tex] square feet.
2. Calculate the total area of the three paintings:
- Since there are three paintings, the total area covered by the paintings is [tex]\( 3 \times 12 = 36 \)[/tex] square feet.
3. Formulate the inequality for the wall area and the combined area:
- The total combined area should be at most 202 square feet. Hence, the inequality becomes:
[tex]\[
\text{Area of the wall} - \text{Area covered by paintings} \leq 202 \text{ square feet}
\][/tex]
4. Express the area of the wall in terms of [tex]\( x \)[/tex]:
- The area of the wall is [tex]\( \text{length} \times \text{width} = (x + 15) \times x = x(x + 15) = x^2 + 15x \)[/tex] square feet.
5. Combine the areas into the inequality:
- Thus, the inequality becomes:
[tex]\[
x^2 + 15x - 36 \leq 202
\][/tex]
6. Isolate the terms on one side of the inequality:
- Subtract 202 from both sides to set the inequality to zero:
[tex]\[
x^2 + 15x - 36 - 202 \leq 0
\][/tex]
- Simplifying this gives:
[tex]\[
x^2 + 15x - 238 \leq 0
\][/tex]
Thus, the correct inequality that can be used to solve for [tex]\( x \)[/tex], given the combined area constraint, is:
[tex]\[ x^2 + 15x - 238 \leq 0 \][/tex]
Therefore, the correct answer is:
A. [tex]\( x^2 + 15x - 238 \leq 0 \)[/tex]
