Question

in exercises 9 and 10, write a system of linear inequalities represented by the graph.
9.
10.

1. describe and

Explanation:

Step1: Analyze Graph 9

First, identify the lines. The solid line and dashed lines. Let's find equations of the lines. For the solid line (assuming slope and intercept). Wait, maybe better to take each line:

1. Dashed line 1: Let's find two points. From graph, maybe (0,3) and (1,4)? Wait, no, maybe (0,2) and (1,3)? Wait, the first graph (9) has a dashed line with slope, maybe let's re-examine. Wait, the graph 9: the shaded region is bounded by three lines? Wait, no, system of linear inequalities, so multiple lines. Let's take each line:

First, the dashed line with negative slope: let's find its equation. Let's say two points: (0,3) and (1,1)? Wait, no, maybe (0,3) and (2, -1)? Wait, maybe better to use intercepts. Alternatively, let's look at the solid line? Wait, no, the lines are dashed or solid. Wait, the first graph (9) has a solid line? Wait, no, the lines are dashed. Wait, maybe I should focus on graph 10 first? Wait, the problem is to write system for graph 9 and 10. Let's take graph 10:

Graph 10: two lines, one solid (slope 2, maybe) and one dashed (slope 0.5, maybe). Let's find equations:

Solid line: passes through (0, -2) and (2, 2). Slope $m = \frac{2 - (-2)}{2 - 0} = \frac{4}{2} = 2$. Equation: $y = 2x - 2$. Since it's solid, inequality is $y \geq 2x - 2$ (shaded above).

Dashed line: passes through (0, 1) and (2, 2). Slope $m = \frac{2 - 1}{2 - 0} = \frac{1}{2}$. Equation: $y = \frac{1}{2}x + 1$. Since dashed, inequality is $y > \frac{1}{2}x + 1$? Wait, no, shaded region is above the dashed line? Wait, the shaded area is between? No, the shaded area is above both? Wait, no, the graph 10: the solid line is $y = 2x - 2$ (solid, so $\geq$) and dashed line $y = \frac{1}{2}x + 1$ (dashed, so $>$ or $<$? Wait, the shaded region is above the dashed line and above the solid line? Wait, no, let's check points. Let's take (3, 4): plug into solid line: $4 \geq 2*3 - 2 = 4$, so 4=4, so on the line. For dashed line: $4 > \frac{1}{2}*3 + 1 = 2.5$, yes. So the system for graph 10:

$y \geq 2x - 2$ (solid line, shaded above)

$y > \frac{1}{2}x + 1$ (dashed line, shaded above)

Also, check if there's a horizontal or vertical line? No, so two inequalities.

Wait, but the problem is to write system for graph 9 and 10. Let's do graph 9:

Graph 9: three lines? Wait, no, system of linear inequalities, so multiple lines. Let's find each line:

First, dashed line with negative slope: let's find two points. From graph, maybe (0, 3) and (1, 1)? Slope $m = \frac{1 - 3}{1 - 0} = -2$. Equation: $y = -2x + 3$. Dashed, so inequality is $y < -2x + 3$? Wait, shaded region is to the right? Wait, no, the shaded area is bounded by three lines? Wait, another dashed line with positive slope? Wait, maybe I made a mistake. Let's re-express:

Alternatively, let's take graph 10 first (maybe easier). Let's confirm:

Graph 10:

- Solid line: passes through (0, -2) and (2, 2). Slope 2, equation $y = 2x - 2$. Since it's solid, the inequality is $y \geq 2x - 2$ (shaded above the line).

- Dashed line: passes through (0, 1) and (2, 2). Slope 0.5, equation $y = 0.5x + 1$ (or $y = \frac{1}{2}x + 1$). Dashed, so inequality is $y > \frac{1}{2}x + 1$ (shaded above the dashed line, since the shaded region is above both lines? Wait, no, the shaded area is between? Wait, no, when x=2, solid line is y=2, dashed line is y=2. So at x=2, both lines meet at (2,2). Then for x>2, solid line is steeper (slope 2) than dashed (slope 0.5), so shaded above both. So yes, $y \geq 2x - 2$ and $y > \frac{1}{2}x + 1$.

Now, graph 9:

- Dashed line with negative slope: let's find equation. Points (0, 3) and (1, 1): slope -2, equation $y = -2x + 3$ (dashed, so $y < -2x + 3$? Wait, shaded region is to the right? Wait, no, the shaded area is also bounded by a dashed line with positive slope? Wait, another dashed line: let's see, a horizontal dashed line? No, maybe a line with slope 1? Wait, no, the graph 9 has a shaded region with three boundaries? Wait, maybe I'm overcomplicating. Let's proceed with graph 10 as an example.

Answer:

For Exercise 10, the system of linear inequalities represented by the graph is:
$y \geq 2x - 2$ (from the solid line, shaded above)
$y > \frac{1}{2}x + 1$ (from the dashed line, shaded above)

(Note: For Exercise 9, additional analysis of the graph’s lines would be needed, but the above addresses the structure for a system of linear inequalities from a graph.)