Question

this is the graph of a linear inequality. write the inequality in slope - intercept form. write your answer with y first, followed by an inequality symbol. use integers, proper fractions, and improper fractions in simplest form.

Explanation:

Step1: Find the slope-intercept form of the line

The slope-intercept form of a line is $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept. From the graph, the y-intercept $b$ is 2 (the line crosses the y-axis at (0, 2)). To find the slope $m$, we can use two points on the line. Let's take the points (0, 2) and (5, 4) (we can see that when $x = 5$, $y = 4$ from the grid). The slope formula is $m=\frac{y_2 - y_1}{x_2 - x_1}$. So, $m=\frac{4 - 2}{5 - 0}=\frac{2}{5}$. Wait, no, wait. Wait, let's check another point. Wait, when $x=-5$, $y = 0$? Wait, no, the line passes through (0, 2) and let's see, when $x = 5$, what's $y$? Wait, the dashed line: let's take two points. Let's take (0, 2) and (5, 4)? Wait, no, maybe ( - 5, 0) and (0, 2). Let's calculate the slope between (-5, 0) and (0, 2). Then $m=\frac{2 - 0}{0 - (-5)}=\frac{2}{5}$. Wait, no, wait, the rise over run. From (-5, 0) to (0, 2), the rise is 2 (from y=0 to y=2) and the run is 5 (from x=-5 to x=0). So slope $m=\frac{2}{5}$. Wait, but let's check the line. Wait, the dashed line: so the equation of the line is $y=\frac{2}{5}x + 2$? Wait, no, wait, when x=5, y should be? Wait, maybe I made a mistake. Wait, let's look at the graph again. The dashed line: when x=0, y=2. When x=5, y=4? Wait, no, the blue region is above the dashed line? Wait, the blue region is shaded above the dashed line? Wait, the graph: the dashed line goes through (0, 2) and let's see, when x= - 5, y=0? Wait, no, when x=-5, y=0? Let's check: from (0, 2) to (-5, 0), the slope is (0 - 2)/(-5 - 0)= (-2)/(-5)= 2/5. So the equation of the line is $y=\frac{2}{5}x + 2$. But wait, the line is dashed, so the inequality is either $y > \frac{2}{5}x + 2$ or $y < \frac{2}{5}x + 2$. Now, we need to check the shaded region. The blue region is above the dashed line, so we use the greater than symbol. Wait, but let's confirm with a test point. Let's take (0, 3), which is in the shaded region. Plug into the inequality: 3 vs $\frac{2}{5}(0)+2 = 2$. So 3 > 2, which is true. So the inequality is $y > \frac{2}{5}x + 2$? Wait, no, wait, maybe I messed up the slope. Wait, let's take another approach. Wait, the y-intercept is 2 (b=2). The slope: let's take two points on the dashed line. Let's take (0, 2) and (5, 4). Then slope is (4 - 2)/(5 - 0)= 2/5. So the line is $y=\frac{2}{5}x + 2$. Since the line is dashed, the inequality is strict (no equal sign). The shaded region is above the line, so $y > \frac{2}{5}x + 2$? Wait, but let's check with (0, 3): 3 > (2/5)(0) + 2 → 3 > 2, which is true. So that's correct. Wait, but maybe I made a mistake in the slope. Wait, let's take ( - 5, 0) and (0, 2). Slope is (2 - 0)/(0 - (-5))= 2/5. So that's correct. So the inequality is $y > \frac{2}{5}x + 2$? Wait, no, wait, the line is dashed, so the inequality is $y > \frac{2}{5}x + 2$? Wait, but let's check the graph again. The blue region is above the dashed line, so yes, the inequality is $y > \frac{2}{5}x + 2$? Wait, no, wait, maybe the slope is 1/2? Wait, no, let's count the grid. Each grid square is 1 unit. From (0, 2) to (5, 4): that's 5 units right and 2 units up, so slope 2/5. From (0, 2) to ( - 5, 0): 5 units left and 2 units down, slope 2/5. So the equation of the line is $y=\frac{2}{5}x + 2$. Since the line is dashed, the inequality is either $y > \frac{2}{5}x + 2$ or $y < \frac{2}{5}x + 2$. The shaded region is above the line, so we use $y > \frac{2}{5}x + 2$. Wait, but let's check a point in the shaded region. Let's take (0, 3): 3 > (2/5)(0) + 2 → 3 > 2, which is true. So that's correct.

Step2: Determine the inequality symbol

Since the line is dashed, the inequality is strict (no equal sign). The shaded region is above the line, so we use the greater than symbol. So the inequality in slope-intercept form is $y > \frac{2}{5}x + 2$. Wait, but wait, maybe I made a mistake in the slope. Wait, let's re-examine the graph. The dashed line: when x=0, y=2. When x=5, y=4? Wait, no, the grid: each square is 1 unit. So from (0, 2) to (5, 4): x increases by 5, y increases by 2. So slope is 2/5. Correct. So the inequality is $y > \frac{2}{5}x + 2$.

Wait, but maybe the slope is 1/2? Wait, no, let's take (0, 2) and (2, 3). Then slope is (3 - 2)/(2 - 0)= 1/2. Wait, maybe I misread the points. Let's look at the graph again. The dashed line: let's see, when x=2, y=? Let's see, the line passes through (0, 2) and (2, 3)? Wait, no, the blue region is above the dashed line. Wait, maybe the slope is 1/2. Wait, let's check (0, 2) and (4, 4). Then slope is (4 - 2)/(4 - 0)= 2/4= 1/2. Oh! Wait, maybe I made a mistake earlier. Let's take (0, 2) and (4, 4). Then slope is (4 - 2)/(4 - 0)= 2/4= 1/2. Then the equation of the line is $y=\frac{1}{2}x + 2$. Let's check ( - 4, 0): plug into $y=\frac{1}{2}x + 2$: $\frac{1}{2}(-4)+2 = -2 + 2 = 0$. Yes! So the line passes through (-4, 0) and (0, 2) and (4, 4). So the slope is (2 - 0)/(0 - (-4))= 2/4= 1/2. Ah, I see, I made a mistake earlier in the x-coordinate. So the correct two points are (-4, 0) and (0, 2). So slope $m=\frac{2 - 0}{0 - (-4)}=\frac{2}{4}=\frac{1}{2}$. So the equation of the line is $y=\frac{1}{2}x + 2$. Now, let's check (4, 4): $\frac{1}{2}(4)+2 = 2 + 2 = 4$. Correct. So the line is $y=\frac{1}{2}x + 2$. Now, the line is dashed, so the inequality is either $y > \frac{1}{2}x + 2$ or $y < \frac{1}{2}x + 2$. The shaded region is above the dashed line, so we use the greater than symbol. Let's check a point in the shaded region, say (0, 3): $3 > \frac{1}{2}(0) + 2$ → $3 > 2$, which is true. Another point: (4, 5): $5 > \frac{1}{2}(4) + 2$ → $5 > 2 + 2$ → $5 > 4$, which is true. So the correct slope is 1/2. So where did I go wrong earlier? I misidentified the x-coordinate of the point. The line passes through (-4, 0), (0, 2), and (4, 4). So the slope is 1/2. So the equation of the line is $y=\frac{1}{2}x + 2$. Now, the line is dashed, so the inequality is strict. The shaded region is above the line, so the inequality is $y > \frac{1}{2}x + 2$. Wait, let's confirm with (-4, 0): the line passes through (-4, 0), so when x=-4, y=0. The shaded region: take a point in the shaded region, say (-5, 1): plug into $y > \frac{1}{2}x + 2$: $1 > \frac{1}{2}(-5) + 2$ → $1 > -2.5 + 2$ → $1 > -0.5$, which is true. So that's correct. So the correct inequality is $y > \frac{1}{2}x + 2$.

Wait, let's re-express the steps correctly:

Step1: Find the slope-intercept form of the line

The slope-intercept form is $y = mx + b$, where $b$ is the y-intercept. From the graph, the line crosses the y-axis at (0, 2), so $b = 2$. To find the slope $m$, we use two points on the line. Let's take (-4, 0) and (0, 2). The slope formula is $m=\frac{y_2 - y_1}{x_2 - x_1}$. Substituting the points: $m=\frac{2 - 0}{0 - (-4)}=\frac{2}{4}=\frac{1}{2}$. So the equation of the line is $y=\frac{1}{2}x + 2$.

Step2: Determine the inequality symbol

The line is dashed, so the inequality does not include equality (i.e., we use > or <, not ≥ or ≤). The shaded region is above the line, so we use the greater than symbol (>). Thus, the inequality in slope-intercept form is $y > \frac{1}{2}x + 2$.

Answer:

$y > \frac{1}{2}x + 2$