Question

graphing linear inequalities guided notes essential question how do i graph linear inequalities and determine its solutions? graphing linear inequalities - to graph a linear inequality, convert the inequality to __. - graph the line with the and . the line will be or based on the inequality sign. a solid line includes solutions on the line, while the dotted line does not. - shade or the line based on which direction makes the inequality __. this can be determined by substituting a test point back into the inequality. solid line dashed line shade above shade below 1. graph $y < \frac{1}{2}x + 4$ 2. graph $3y \geq -x - 6$ 3. graph $3x - 5y \geq 15$ 4. graph $2x + 3y > 9$ 5. graph $x < 2$ 6. graph $y \geq -3$

Explanation:

Step1: Analyze the inequality \( y < \frac{1}{2}x + 4 \)

First, identify the boundary line. The inequality is in slope - intercept form \( y=mx + b \), where \( m=\frac{1}{2} \) (slope) and \( b = 4 \) (y - intercept). Since the inequality sign is \( < \), the boundary line will be dashed (because the inequality is strict, points on the line are not included in the solution set).

Step2: Graph the boundary line

Plot the y - intercept at \( (0,4) \). Then, use the slope \( \frac{1}{2} \) (rise over run, so from \( (0,4) \), move up 1 unit and right 2 units to get the next point \( (2,5) \), or down 1 unit and left 2 units to get \( (- 2,3) \)). Draw a dashed line through these points.

Step3: Determine the shading region

Choose a test point that is not on the line. A common test point is \( (0,0) \). Substitute \( x = 0 \) and \( y = 0 \) into the inequality \( y<\frac{1}{2}x + 4 \):
\( 0<\frac{1}{2}(0)+4 \)
\( 0 < 4 \), which is a true statement. So, we shade the region that contains the test point \( (0,0) \), which is below the line \( y=\frac{1}{2}x + 4 \).

Answer:

To graph \( y < \frac{1}{2}x + 4 \):

1. Boundary Line: Dashed line with slope \( \frac{1}{2} \) and y - intercept \( 4 \) (equation \( y=\frac{1}{2}x + 4 \)).
2. Shading: Shade the region below the dashed line (the region containing \( (0,0) \) as the test point satisfies the inequality).