Question

1. a carpenter has at most $250 to spend on lumber. the inequality 8x + 12y ≤ 250 represents the numbers x of 2-by-8 boards and the numbers y of 4-by-4 boards the carpenter can buy. can the carpenter buy twelve 2-by-8 boards and fourteen 4-by-4 boards?

Explanation:

Step1: Identify x and y values

Here, \( x = 12 \) (2 - by - 8 boards) and \( y=14 \) (4 - by - 4 boards).

Step2: Substitute into inequality

Substitute \( x = 12 \) and \( y = 14 \) into the left - hand side of the inequality \( 8x + 12y\leq250 \).
First, calculate \( 8x \): \( 8\times12=96 \).
Then, calculate \( 12y \): \( 12\times14 = 168 \).
Now, find the sum: \( 8x+12y=96 + 168=264 \).

Step3: Compare with 250

We compare \( 264 \) with \( 250 \). Since \( 264>250 \), the inequality \( 8x + 12y\leq250 \) is not satisfied.

Answer:

No, the carpenter cannot buy twelve 2 - by - 8 boards and fourteen 4 - by - 4 boards because when we substitute \( x = 12 \) and \( y = 14 \) into the inequality \( 8x+12y\leq250 \), we get \( 8\times12 + 12\times14=96 + 168 = 264\), and \( 264>250 \).