determine the inequality shown in the graph below. the inequality symbols can be entered using the toolbar.

determine the inequality shown in the graph below. the inequality symbols can be entered using the toolbar.

determine the inequality shown in the graph below. the inequality symbols can be entered using the toolbar.

Answer

Explanation:

Step1: Find the equation of the boundary line

The boundary line is a straight line. Let's find two points on the line. From the graph, when (x = 0), (y=- 2) (approximate) and when (y = 5), (x = 2) (approximate). Wait, actually, let's calculate the slope. Let's take two clear points. Looking at the grid, when (x = 1), (y = 3)? Wait, maybe better to see the slope. Let's see, the line passes through ((0,-2)) and ((2,4))? Wait, no, let's check the slope. The line has a slope (m=\frac{y_2 - y_1}{x_2 - x_1}). Let's take two points: when (x = 0), (y=-2) and when (x = 2), (y = 4). So (m=\frac{4-\left(-2\right)}{2 - 0}=\frac{6}{2}=3). So the equation of the line is (y=3x - 2) (using slope - intercept form (y=mx + b), where (b=-2) when (x = 0), (y=-2)).

Step2: Determine the inequality symbol

The boundary line is dashed (since it's a red dashed line), so the inequality is strict (either (>) or (<)). The shaded region is to the left of the line. Let's test a point in the shaded region, say ((0,0)). Plug into (y) and (3x - 2): (0) vs (3(0)-2=-2). Since (0>-2)? Wait, no, the shaded region is on the left. Wait, maybe I made a mistake in the line equation. Let's re - examine the graph. Let's take two points on the dashed line. Let's see, when (x = 1), (y = 1)? No, maybe the slope is (2). Wait, let's take ((0,-2)) and ((1,0)). Then slope (m=\frac{0-\left(-2\right)}{1 - 0}=2). So equation (y = 2x-2). Test point ((0,0)): (0) vs (2(0)-2=-2). (0>-2), but the shaded region is to the left of the line. Wait, maybe the line is (y=2x - 2), and the shaded region is (y<2x - 2)? No, wait, the shaded area is on the side where for a given (x), (y) is less than the line? Wait, no, let's look at the graph again. The dashed line goes from the bottom left to the top right. The shaded region is to the left of the line. Let's take a point in the shaded region, say ((-5,0)). Plug into (y) and (2x - 2): (0) vs (2(-5)-2=-12). (0>-12), but the shaded region is where (y<2x - 2)? No, that's not right. Wait, maybe the line is (y = 2x-2), and the inequality is (y<2x - 2)? No, let's think differently. The general form of a linear inequality with a dashed line (ax+by + c = 0) has the inequality (ax + by + c>0) or (ax + by + c<0). Let's write the line as (2x-y-2 = 0) (from (y = 2x-2) we get (2x-y-2=0)). Now, test the point ((0,0)) in (2x - y-2): (2(0)-0 - 2=-2<0). The shaded region: if we take ((-5,0)), (2(-5)-0 - 2=-12<0). Wait, no, the shaded region is where (2x - y-2<0), which is equivalent to (y>2x - 2)? No, (2x - y-2<0) implies (y>2x - 2)? Wait, (2x-y-2<0) can be rewritten as (-y<-2x + 2), then (y>2x - 2) (multiplying both sides by - 1 reverses the inequality). But the shaded region is to the left of the line, so maybe the correct line is (y = 2x-2) and the inequality is (y<2x - 2)? I think I messed up the direction. Let's start over.

Wait, the standard way:

  1. Find the equation of the boundary line:
    • The boundary line is a straight line. Let's find two points on the line. From the graph, when (x = 0), (y=-2) and when (x = 1), (y = 0). So the slope (m=\frac{0-\left(-2\right)}{1 - 0}=2). So the equation of the line is (y = 2x-2) (using (y=mx + b), (b=-2)).
  2. Determine the inequality:
    • The line is dashed, so the inequality is strict ((>) or (<)).
    • The shaded region is below or above? Wait, the shaded region is on the side where (y) is less than the line? Wait, no, let's take a point in the shaded area, say ((0,0)). Plug into (y) and (2x - 2): (0) vs (2(0)-2=-2). Since (0>-2), but the shaded region is to the left of the line. Wait, maybe the correct inequality is (y<2x - 2)? No, when (x = 0), (y=-2), and the shaded region at (x = 0) has (y) values from (-5) to (5) (the shaded area at (x = 0) is above and below? No, the graph shows a triangular - like shaded region? Wait, no, the graph is a region bounded by the dashed line, the (x) - axis, and (y) - axis? No, the graph is a region with a dashed line and shaded to the left.

Wait, maybe the line is (y = 3x-2), and the inequality is (y<3x - 2)? No, let's use the correct method.

The general steps for finding the inequality of a linear graph:

  1. Identify the equation of the boundary line:
    • If the line is dashed, it's a strict inequality ((>) or (<)); if solid, it's non - strict ((\geq) or (\leq)).
    • Find two points on the line. Let's assume the line passes through ((0,-2)) and ((1,1)). Then slope (m=\frac{1-\left(-2\right)}{1 - 0}=3), so equation (y = 3x-2).
  2. Test a point in the shaded region:
    • Let's take the point ((-1,0)) (in the shaded region). Plug into (y) and (3x - 2): (0) vs (3(-1)-2=-5). Since (0>-5), but the shaded region is on the side where (y<3x - 2)? No, (0>-5) means (y>3x - 2) at ((-1,0)). But the shaded region is to the left of the line. Wait, maybe the inequality is (y<3x - 2) is wrong. Wait, the correct approach is:

The boundary line is (y = 2x-2) (let's re - check with two points: when (x = 1), (y = 0); when (x = 2), (y = 2), slope (m = 2), equation (y=2x - 2)).

Test the point ((0,0)) in the inequality:

For (y<2x - 2): (0<2(0)-2=-2), which is false.

For (y>2x - 2): (0>2(0)-2=-2), which is true. But the shaded region is to the left of the line. Wait, maybe the inequality is (x<\frac{y + 2}{2}), but that's not standard.

Wait, maybe I made a mistake in the line equation. Let's look at the graph again. The dashed line intersects the (y) - axis at ((0,-2)) and the (x) - axis at ((1,0)). So the slope is (m=\frac{0-\left(-2\right)}{1 - 0}=2), so the equation is (y = 2x-2). The shaded region is where (y<2x - 2)? No, when (x = 0), the shaded region includes (y) values from (-5) to (5), and (2x-2=-2) at (x = 0). So for (x = 0), (y) in the shaded region is less than (-2)? No, the shaded region at (x = 0) is above and below? No, the graph shows a region that is to the left of the dashed line.

Wait, another way: the inequality can be written in terms of (x) and (y). The line is (y = 2x-2), or (2x-y-2 = 0). The shaded region is on the side where (2x-y-2>0) (because when we solve (2x-y-2>0) for (y), we get (y<2x - 2)? No, (2x-y-2>0) implies (y<2x - 2)). Wait, no: (2x-y-2>0) (\Rightarrow -y>-2x + 2) (\Rightarrow y<2x - 2) (multiplying both sides by - 1 reverses the inequality).

Let's test the point ((-5,0)) in (2x-y-2): (2(-5)-0 - 2=-12<0), which would mean (y>2x - 2) (since (2x-y-2<0) (\Rightarrow y>2x - 2)). Ah! Here is the mistake. If (2x-y-2<0), then (y>2x - 2). Let's test ((-5,0)): (0>2(-5)-2=-12), which is true. And ((0,0)): (0>2(0)-2=-2), which is true. And the boundary line is dashed, so the inequality is (y>2x - 2)? But the shaded region is to the left of the line. Wait, maybe the correct inequality is (y<2x - 2) is wrong, and it's (y>2x - 2) with the shaded region on the left.

Wait, maybe the line is (y = 3x-2). Let's test ((-5,0)) in (y = 3x-2): (0=3(-5)-2=-17)? No, that's not. I think I need to re - evaluate.

Wait, the key steps are:

  1. Find the equation of the boundary line:
    • The line is dashed, so it's a linear equation with a strict inequality.
    • From the graph, the line has a slope of (2) (rise over run: from ((0,-2)) to ((1,0)), rise is (2), run is (1)) and (y) - intercept of (-2), so (y = 2x-2).
  2. Determine the inequality:
    • The shaded region is where (y<2x - 2) is incorrect. Wait, no, let's take a point in the shaded area, say ((-2,0)). Plug into (y) and (2x - 2): (0) vs (2(-2)-2=-6). Since (0>-6), the inequality is (y>2x - 2). But the boundary line is dashed, so the inequality is (y>2x - 2).

Wait, but the graph's shaded region is to the left of the line. Let's think in terms of (x). For a given (y), (x) in the shaded region is less than (\frac{y + 2}{2}). Which is equivalent to (2x<y + 2) (\Rightarrow 2x-y-2<0) (\Rightarrow y>2x - 2). Yes, that's correct. So the inequality is (y>2x - 2).

Wait, but maybe I made a mistake in the slope. Let's check with another pair of points. If the line passes through ((0,-2)) and ((2,2)), slope (m=\frac{2-\left(-2\right)}{2 - 0}=2), so equation (y = 2x-2) is correct.

So the inequality is (y>2x - 2) (since the line is dashed and the shaded region is where (y) is greater than the line's (y) - value for a given (x)).

Answer:

(y>2x - 2)