Question

4-6. write an inequality that represents (and then there is a coordinate grid with a dashed line)

Explanation:

Step1: Find the slope of the dashed line

The dashed line passes through points, let's take two points: when \( x = 1 \), \( y = 0 \) (approx) and when \( x = 0 \), \( y=-5 \) (approx). Wait, better to calculate slope \( m=\frac{y_2 - y_1}{x_2 - x_1} \). Let's find two clear points. Looking at the grid, when \( x = 1 \), \( y = 0 \); when \( x = 0 \), \( y=-5 \)? Wait, no, maybe another pair. Wait, the line seems to have a slope. Let's take two points: (1, 0) and (0, -5)? Wait, no, maybe (1, 0) and (2, 5)? Wait, no, the grid lines: each square is 1 unit. Let's see, the dashed line: when \( x = 1 \), \( y = 0 \); when \( x = 0 \), \( y=-5 \)? No, that can't be. Wait, maybe I made a mistake. Wait, the line is dashed, so it's a linear inequality. Let's find the equation of the dashed line first. Let's take two points on the dashed line. Let's see, when \( x = 1 \), \( y = 0 \); when \( x = 0 \), \( y=-5 \)? No, that's not right. Wait, maybe the line passes through (1, 0) and (0, -5)? Wait, no, let's count the rise over run. From (0, -5) to (1, 0): rise is 5, run is 1, so slope \( m = 5 \)? Wait, no, that seems steep. Wait, maybe another approach. Let's take the general form of a line: \( y = mx + b \). Let's find \( m \) and \( b \). Let's pick two points on the dashed line. Let's see, when \( x = 1 \), \( y = 0 \); when \( x = 2 \), \( y = 5 \)? No, that's not matching. Wait, maybe the line is \( y = 5x - 5 \)? Wait, when \( x = 1 \), \( y = 0 \); when \( x = 0 \), \( y=-5 \). Yes, that works. So the equation of the dashed line is \( y = 5x - 5 \). Now, the shaded region: the shaded area is to the right of the dashed line? Wait, no, looking at the graph, the shaded region is on the right side of the dashed line. Wait, let's test a point in the shaded region, say (2, 0). Plug into \( y = 5x - 5 \): \( 0 = 10 - 5 = 5 \)? No, that's not. Wait, maybe I got the slope wrong. Let's try again. Let's take two points: (1, 0) and (0, -5). Slope \( m=\frac{0 - (-5)}{1 - 0}=5 \). So equation is \( y = 5x - 5 \). Now, the shaded region: let's pick a point in the shaded region, say (2, 0). Plug into \( y < 5x - 5 \)? \( 0 < 10 - 5 = 5 \), which is true. Wait, but the dashed line means the inequality is strict (no equal sign). Wait, but maybe the slope is different. Wait, maybe I made a mistake in the points. Let's look again. The grid: x-axis from -5 to 5, y-axis from -8 to 2. The dashed line: when x=1, y=0; x=0, y=-5; x=2, y=5. So the slope is 5. Now, the inequality: the shaded region is to the right of the line? Wait, no, when x=2, y=0 is in the shaded region? Wait, the shaded area is the region with the dashed line, and the shaded part is on the right. Wait, let's test the point (2, 0) in the inequality. If the line is \( y = 5x - 5 \), then for (2, 0): \( 0 \) vs \( 5*2 -5 = 5 \). So \( 0 < 5 \), so the inequality would be \( y < 5x - 5 \)? But that seems off. Wait, maybe the slope is 5, but the equation is \( y = 5x - 5 \), and the shaded region is \( y > 5x - 5 \)? Wait, no, let's take a point in the shaded region, say (2, 3). Plug into \( y = 5x - 5 \): \( 3 = 10 - 5 = 5 \)? No, 3 < 5. Wait, maybe the slope is 5, and the inequality is \( x > \frac{y + 5}{5} \), but that's the same as \( y < 5x - 5 \). Wait, maybe I messed up the direction. Alternatively, let's use the standard method: find the equation of the dashed line, then determine the inequality.

Wait, maybe the line is \( y = 5x - 5 \), and the shaded region is \( y < 5x - 5 \)? But let's check the point (0,0). Plug into \( y < 5x - 5 \): \( 0 < -5 \), which is false. But (0,0) is not in the shaded region? Wait, the shaded region is on the right of the dashed line. Wait, (0,0): x=0, y=0. The dashed line at x=0 is y=-5, so (0,0) is above the dashed line at x=0. Wait, maybe the line is \( y = 5x - 5 \), and the shaded region is \( x > \frac{y + 5}{5} \), but that's complicated. Wait, maybe the slope is 5, and the inequality is \( y < 5x - 5 \), but let's check another point. Take (2, 5): plug into \( y < 5x - 5 \): \( 5 < 10 - 5 = 5 \)? No, 5 is not less than 5. So that's not right. Wait, maybe the slope is 5, but the equation is \( y = 5x - 5 \), and the inequality is \( y > 5x - 5 \). Then (2,5): \( 5 > 10 - 5 = 5 \)? No, 5 is not greater than 5. Hmm, maybe I made a mistake in the slope. Let's try again. Let's take two points on the dashed line: (1, 0) and (0, -5). Slope is (0 - (-5))/(1 - 0) = 5/1 = 5. So equation is \( y = 5x - 5 \). Now, the shaded region: is it above or below? Wait, the dashed line is the boundary. Let's take a point in the shaded region, say (2, 0). Plug into \( y = 5x - 5 \): 0 = 10 - 5 = 5? No, 0 < 5. So if the shaded region is where \( y < 5x - 5 \), but (2,0) is in the shaded region, and 0 < 5 is true. Wait, but (0,0): 0 < -5? No, false, and (0,0) is not in the shaded region. That makes sense. So the inequality is \( y < 5x - 5 \), but let's write it in standard form. \( 5x - y - 5 > 0 \) or \( 5x - y > 5 \). Wait, maybe I made a mistake in the y-intercept. Let's check again. When x=1, y=0: 0 = 5(1) + b => b = -5. So equation is \( y = 5x - 5 \). So the inequality, since the line is dashed (so no equal sign) and the shaded region is where \( y < 5x - 5 \) (because (2,0) is in shaded and 0 < 5*2 -5 = 5). So the inequality is \( y < 5x - 5 \) or \( 5x - y > 5 \).

Wait, but maybe the slope is different. Let's take another pair of points. Let's see, when x=1, y=0; x=2, y=5. So from (1,0) to (2,5): rise is 5, run is 1, slope 5. Correct. So the equation of the dashed line is \( y = 5x - 5 \), and the shaded region is \( y < 5x - 5 \) (since the shaded area is below the line? Wait, no, (2,0) is below the line \( y = 5x - 5 \) (since 0 < 5). So yes, the inequality is \( y < 5x - 5 \) or \( 5x - y > 5 \).

Step2: Determine the inequality sign

Since the line is dashed, the inequality is strict (no \( \leq \) or \( \geq \)). The shaded region is where the inequality holds. Let's test a point in the shaded region, say (2, 0). Plug into \( y < 5x - 5 \): \( 0 < 5(2) - 5 = 5 \), which is true. Plug into \( y > 5x - 5 \): \( 0 > 5 \), which is false. So the inequality is \( y < 5x - 5 \) or rearranged as \( 5x - y > 5 \).

Answer:

The inequality representing the graph is \( \boldsymbol{y < 5x - 5} \) (or \( \boldsymbol{5x - y > 5} \))