Question

1. one large box and two small boxes weigh a total of 50 pounds. two large boxes and three small boxes weigh a total of 90 pounds. the weight limit of a trailer is 300 pounds. write and graph an inequality that represents the numbers of large and small boxes that can be loaded in the trailer.

Explanation:

Step1: Define Variables

Let \( x \) be the number of large boxes and \( y \) be the number of small boxes. Let \( L \) be the weight of one large box and \( S \) be the weight of one small box.

From the problem:

1. \( L + 2S = 50 \)
2. \( 2L + 3S = 90 \)

Step2: Solve for \( L \) and \( S \)

From the first equation, express \( L \) in terms of \( S \): \( L = 50 - 2S \)

Substitute into the second equation:
\( 2(50 - 2S) + 3S = 90 \)
\( 100 - 4S + 3S = 90 \)
\( 100 - S = 90 \)
\( -S = -10 \)
\( S = 10 \)

Then, \( L = 50 - 2(10) = 30 \)

Step3: Write the Inequality

The total weight of \( x \) large boxes and \( y \) small boxes must be less than or equal to 300 pounds. So:
\( 30x + 10y \leq 300 \)
Simplify by dividing by 10:
\( 3x + y \leq 30 \)

Step4: Graph the Inequality

1. Rewrite in slope - intercept form: \( y \leq - 3x + 30 \)
2. The boundary line is \( y=-3x + 30 \), which is a straight line with slope - 3 and y - intercept 30.
3. Since the inequality is \( y\leq - 3x + 30 \), we shade the region below the line (including the line itself because of the "less than or equal to" sign). The x - intercept of the line \( y=-3x + 30 \) is found by setting \( y = 0 \): \( 0=-3x + 30\Rightarrow x = 10 \), and the y - intercept is 30. We can plot the points (0, 30) and (10, 0) and draw the line through them, then shade the region below the line.

Answer:

The inequality is \( 3x + y\leq30 \) (or \( 30x + 10y\leq300 \)). To graph it, draw the line \( y=-3x + 30 \) (a straight line with slope - 3, y - intercept 30, and x - intercept 10) and shade the region below the line (including the line).