Question

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graph each system of inequalities to show all possible solutions.

1. \

$$\begin{cases} y > -2x + 5 \\\\ y \\leq x - 3 \\end{cases}$$

1. \

$$\begin{cases} y \\leq -\\frac{2}{3}x - 3 \\\\ y \\geq 3x + 8 \\end{cases}$$

1. \

$$\begin{cases} 5x - 6y > 42 \\\\ 3x + 2y \\leq 14 \\end{cases}$$

1. \

$$\begin{cases} 7x + 3y \\leq -24 \\\\ x + 3y \\leq -6 \\end{cases}$$

1. \

$$\begin{cases} x - y > 3 \\\\ x \\geq 6 \\end{cases}$$

1. \

$$\begin{cases} y < 2x + 5 \\\\ 2x - y \\leq 3 \\end{cases}$$

© gina wilson (all things algebra)

Explanation:

Step1: Analyze \( y > -2x + 5 \)

The inequality \( y > -2x + 5 \) is a linear inequality. The boundary line is \( y = -2x + 5 \), which has a slope of \( -2 \) and a y - intercept of \( 5 \). Since the inequality is \( y > \) (not \( \geq \)), the boundary line should be dashed. To determine the region to shade, we can test a point not on the line, say \( (0,0) \). Substituting into the inequality: \( 0 > -2(0)+5 \) which simplifies to \( 0 > 5 \), a false statement. So we shade the region above the dashed line \( y = -2x + 5 \).

Step2: Analyze \( y \leq x - 3 \)

The inequality \( y \leq x - 3 \) has a boundary line \( y = x - 3 \) with a slope of \( 1 \) and a y - intercept of \( -3 \). Since the inequality is \( \leq \), the boundary line is solid. Testing the point \( (0,0) \): \( 0 \leq 0 - 3 \) simplifies to \( 0 \leq -3 \), a false statement. So we shade the region below the solid line \( y = x - 3 \).

Step3: Find the Intersection Region

The solution to the system of inequalities is the region that is shaded for both inequalities. So we need to find the area that is above the dashed line \( y=-2x + 5 \) and below the solid line \( y = x - 3 \). To graph this, first draw the dashed line \( y=-2x + 5 \) (passing through \( (0,5) \) and with a slope of - 2, so another point is \( (1,3) \)) and shade above it. Then draw the solid line \( y=x - 3 \) (passing through \( (0,-3) \) and with a slope of 1, so another point is \( (1,-2) \)) and shade below it. The overlapping region (the solution set) is where these two shaded regions intersect.

(Note: Since the problem is about graphing, the final answer is the graph with the dashed line \( y=-2x + 5 \), solid line \( y = x - 3 \), and the overlapping shaded region as described above. If we were to describe the graph in terms of the given grid, we can see from the original student's work that the red line is \( y=x - 3 \) and the blue line is \( y=-2x + 5 \), and the solution region is the area that is above the blue dashed line and below the red solid line.)

Answer:

The graph has a dashed line \( y = - 2x+5 \) (slope - 2, y - intercept 5) and a solid line \( y=x - 3 \) (slope 1, y - intercept - 3). The solution region is shaded above the dashed line and below the solid line (the overlapping region of the two shaded areas for each inequality).