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 line passes through \( (0, 5) \) (so \( b = 5 \)) and \( (6, 0) \). The slope \( m \) is calculated as \( \frac{y_2 - y_1}{x_2 - x_1} \). Using the points \( (0, 5) \) and \( (6, 0) \), we have \( m=\frac{0 - 5}{6 - 0}=-\frac{5}{6} \). So the equation of the line is \( y = -\frac{5}{6}x + 5 \).

Step2: Determine the inequality symbol

The line is solid (since the boundary is included, we'll check the shading). The shading is above the line (since the blue region is above the line). For a linear inequality, if the shading is above the line \( y = mx + b \), the inequality is \( y\geq mx + b \) (if the line is solid) or \( y>mx + b \) (if the line is dashed). Here, the line is solid (from the graph, the boundary is a solid line), so we use \( \geq \) or \( \leq \)? Wait, wait, let's check the shading. Wait, the blue region: let's take a test point, say \( (0, 6) \), which is in the blue region. Plug into the line equation \( y = -\frac{5}{6}x + 5 \). At \( x = 0 \), \( y = 5 \). The point \( (0, 6) \) has \( y = 6 \), which is greater than 5. So the inequality is \( y\geq -\frac{5}{6}x + 5 \)? Wait, no, wait the line: let's re - check the points. Wait, the y - intercept is 5 (when \( x = 0 \), \( y = 5 \)), and the x - intercept is 6 (when \( y = 0 \), \( x = 6 \)). So the slope \( m=\frac{0 - 5}{6 - 0}=-\frac{5}{6} \). Now, the line is solid, so the inequality includes equality. Now, to determine the direction of the inequality: we pick a point in the shaded region. Let's take \( (0, 6) \). Plug into the left - hand side and right - hand side of the inequality. If the line is \( y=-\frac{5}{6}x + 5 \), then for \( (0, 6) \), \( 6\) compared to \( -\frac{5}{6}(0)+5 = 5 \). Since \( 6>5 \), the inequality is \( y\geq -\frac{5}{6}x + 5 \)? Wait, no, wait the line: wait, maybe I made a mistake in the slope. Wait, from \( (0,5) \) to \( (6,0) \), the change in \( y \) is \( 0 - 5=-5 \), change in \( x \) is \( 6 - 0 = 6 \), so slope \( m=\frac{-5}{6}=-\frac{5}{6} \). Now, the shaded region: looking at the graph, the blue region is above the line? Wait, no, when \( x = 0 \), the line is at \( y = 5 \), and the shaded region is above \( y = 5 \) (since at \( x = 0 \), the shaded region includes \( y = 6,7,8 \) etc.). So when we test \( (0,6) \), \( 6>5 \), so the inequality is \( y\geq -\frac{5}{6}x + 5 \)? Wait, but let's check another point, say \( (6,0) \) is on the line. What about \( (0,5) \), which is on the line. Now, let's check the line type: the line is solid, so the inequality is either \( y\geq mx + b \) or \( y\leq mx + b \). Let's take a point in the non - shaded region, say \( (0,4) \). \( 4<5 \), and \( (0,4) \) is not in the shaded region. So the shaded region is where \( y\geq -\frac{5}{6}x + 5 \). Wait, but wait, the slope: wait, maybe I got the slope wrong. Wait, from \( (0,5) \) to \( (6,0) \), the slope is \( \frac{0 - 5}{6 - 0}=-\frac{5}{6} \), that's correct. So the equation of the line is \( y = -\frac{5}{6}x + 5 \), and since the line is solid and the shading is above the line, the inequality is \( y\geq -\frac{5}{6}x + 5 \)? Wait, no, wait, maybe I mixed up. Wait, let's think again. The slope - intercept form is \( y=mx + b \). We have \( b = 5 \), \( m=-\frac{5}{6} \). Now, the line is solid, so the inequality is either \( y\leq mx + b \) or \( y\geq mx + b \). Let's take the point \( (6,0) \) (on the line) and \( (0,5) \) (on the line). Now, take a point in the shaded region, say \( (0,6) \): \( 6\) vs \( -\frac{5}{6}(0)+5 = 5 \). Since \( 6>5 \), the inequality is \( y\geq -\frac{5}{6}x + 5 \). Wait, but let's check another point, say \( ( - 6,10) \). Plug into the line equation: \( y=-\frac{5}{6}(-6)+5=5 + 5 = 10 \). Wait, \( ( - 6,10) \) is on the line? Wait, no, \( -\frac{5}{6}(-6)=5 \), \( 5 + 5 = 10 \), so \( ( - 6,10) \) is on the line. Wait, maybe the line is \( y=-\frac{5}{6}x + 5 \), and the shaded region is above the line, so the inequality is \( y\geq -\frac{5}{6}x + 5 \).

Wait, no, I think I made a mistake. Wait, the line: from \( (0,5) \) to \( (6,0) \), the slope is \( -\frac{5}{6} \). Now, let's check the direction of the inequality. Let's take the origin \( (0,0) \). Is \( (0,0) \) in the shaded region? No, because the shaded region is blue, and at \( x = 0,y = 0 \), it's below the line \( y = -\frac{5}{6}x+5 \) (since \( 0<5 \)). So the shaded region is where \( y\) is greater than or equal to \( -\frac{5}{6}x + 5 \). So the inequality is \( y\geq -\frac{5}{6}x + 5 \). Wait, but let's re - express the line equation correctly. The slope \( m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{0 - 5}{6 - 0}=-\frac{5}{6} \), y - intercept \( b = 5 \), so the line is \( y=-\frac{5}{6}x + 5 \). Since the line is solid (so the inequality includes equality) and the shaded region is above the line (as tested by \( (0,6) \)), the inequality is \( y\geq -\frac{5}{6}x + 5 \).

Wait, no, wait a second. Wait, when \( x = 6 \), the line is at \( y = 0 \), and the shaded region at \( x = 6 \) is above \( y = 0 \) (since the blue region at \( x = 6 \) is above \( y = 0 \)). So yes, the inequality is \( y\geq -\frac{5}{6}x + 5 \).

Answer:

\( y\geq-\frac{5}{6}x + 5 \)