Question

item a takes 10 hours of work to produce. item b takes 8 hours. employees work at most 40 hours a week. in the inequalities below, x represents the number of item a made and y represents the number of item b made. graph the intersection of these inequalities.\

$$\begin{cases}x\\geq 0\\\\y\\geq 0\\\\10x + 8y\\leq 40\\end{cases}$$

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use the graphing tool to graph the intersection.\
click to enlarge graph

Explanation:

Step1: Analyze \( x \geq 0 \) and \( y \geq 0 \)

The inequalities \( x \geq 0 \) and \( y \geq 0 \) mean we are working in the first quadrant (where both \( x \) and \( y \) are non - negative).

Step2: Analyze \( 10x + 8y \leq 40 \)

First, rewrite the inequality as an equation to find the boundary line: \( 10x+8y = 40 \).
We can find the x - intercept by setting \( y = 0 \):
\( 10x+8(0)=40\)
\( 10x=40\)
\( x = 4 \). So the x - intercept is \( (4,0) \).
We can find the y - intercept by setting \( x = 0 \):
\( 10(0)+8y=40\)
\( 8y = 40\)
\( y = 5 \). So the y - intercept is \( (0,5) \).
Since the inequality is \( 10x + 8y\leq40 \) (the "less than or equal to" sign), we draw a solid line through \( (4,0) \) and \( (0,5) \), and shade the region below the line (because when we test the origin \( (0,0) \): \( 10(0)+8(0)=0\leq40 \), which is true, so the origin is in the shaded region).

Step3: Find the intersection

The intersection of \( x\geq0 \), \( y\geq0 \), and \( 10x + 8y\leq40 \) is the region in the first quadrant that is also below (or on) the line \( 10x + 8y = 40 \). This region is a polygon (a triangle in this case) with vertices at \( (0,0) \), \( (4,0) \), and \( (0,5) \). To graph it, we:

1. Draw the x - axis and y - axis.
2. Plot the points \( (0,0) \), \( (4,0) \), and \( (0,5) \).
3. Draw a solid line between \( (4,0) \) and \( (0,5) \).
4. Shade the region that is in the first quadrant (where \( x\geq0 \) and \( y\geq0 \)) and below the line \( 10x + 8y = 40 \).

Answer:

The intersection is the region in the first quadrant bounded by the x - axis, y - axis, and the line \( 10x + 8y = 40 \) (with vertices at \((0,0)\), \((4,0)\), and \((0,5)\)). When graphing, use a solid line for \( 10x + 8y = 40 \), shade the area in the first quadrant below this line.