Question

y < -3x + 4

Explanation:

Answer:

To graph \( y < -3x + 4 \):

Step1: Graph the boundary line

The inequality is \( y < -3x + 4 \). The boundary line is \( y=-3x + 4 \) (in slope - intercept form \( y = mx + b \), where \( m=-3 \) (slope) and \( b = 4 \) (y - intercept)). Since the inequality is \( y < -3x+4 \) (not \( y\leq - 3x + 4 \)), we draw a dashed line.

• For the y - intercept, when \( x = 0 \), \( y=4 \), so the line passes through the point \( (0,4) \).
• Using the slope \( m=-3=\frac{-3}{1} \), from the point \( (0,4) \), we can find another point by moving 1 unit to the right (increase \( x \) by 1) and 3 units down (decrease \( y \) by 3). So when \( x = 1 \), \( y=4-3 = 1 \), so the line also passes through \( (1,1) \). We can draw the dashed line through these two (and other) points.

Step2: Shade the region

To determine which side of the line to shade, we can test a point that is not on the line. A common test point is the origin \( (0,0) \) (since it is easy to substitute into the inequality).
Substitute \( x = 0 \) and \( y = 0 \) into the inequality \( y < -3x+4 \):
\( 0< - 3(0)+4 \)
\( 0 < 4 \), which is a true statement.
Since the origin satisfies the inequality, we shade the region that contains the origin (the region below the dashed line \( y=-3x + 4 \)).

(Note: If you were to actually draw the graph, the dashed line would go through (0,4) and (1,1) (and other points with the slope - 3), and the area below the line (including all points where the y - value is less than \( - 3x + 4 \)) would be shaded.)