Question

1. graph the system of inequalities:

y ≤ x - 4
3x + y ≤ 4

create two tables! plot the points!

Explanation:

for \( y \leq x - 4 \):

Step 1: Rewrite as equation

First, consider the boundary line \( y = x - 4 \) (a linear equation in slope - intercept form \( y=mx + b \), where \( m = 1 \) and \( b=-4 \)).

Step 2: Create a table of values

We can choose some values for \( x \) and calculate the corresponding \( y \) values.

\( x \)	\( y=x - 4 \)
\( 4 \)	\( y = 4-4 = 0 \)
\( 5 \)	\( y=5 - 4 = 1 \)

Since the inequality is \( y\leq x - 4 \), the line will be solid (because of the "less than or equal to" sign), and we will shade the region below the line. To test the region, we can use a test point, for example, the point \( (0,0) \). Substitute into the inequality: \( 0\leq0 - 4\)? \( 0\leq - 4\) is false, so we shade the region that does not contain \( (0,0) \), which is below the line \( y=x - 4 \).

for \( 3x + y\leq4 \):

Step 1: Rewrite as equation

Rewrite the inequality \( 3x + y\leq4 \) as the equation \( y=-3x + 4 \) (slope - intercept form, where \( m=-3 \) and \( b = 4 \)).

Step 2: Create a table of values

Choose values for \( x \) and find \( y \):

\( x \)	\( y=-3x + 4 \)
\( 1 \)	\( y=-3(1)+4 = 1 \)
\( 2 \)	\( y=-3(2)+4=-2 \)

The boundary line is solid (because of the "less than or equal to" sign). Use a test point, say \( (0,0) \). Substitute into the inequality: \( 3(0)+0\leq4 \)? \( 0\leq4 \) is true. So we shade the region that contains \( (0,0) \), which is below the line \( y = - 3x+4 \).

Plotting the Points:

• For the line \( y=x - 4 \), plot the points from its table (\( (0,-4) \), \( (4,0) \), \( (5,1) \)) and draw a solid line through them. Then shade the region below the line.
• For the line \( 3x + y=4 \) (or \( y=-3x + 4 \)), plot the points from its table (\( (0,4) \), \( (1,1) \), \( (2,-2) \)) and draw a solid line through them. Then shade the region below the line.

The solution to the system of inequalities is the region where the two shaded regions overlap.

Answer:

To graph the system:

1. For \( y\leq x - 4 \):

• Draw the solid line \( y = x-4 \) using the points from its table.
• Shade the region below the line.

1. For \( 3x + y\leq4 \):

• Draw the solid line \( y=-3x + 4 \) using the points from its table.
• Shade the region below the line.

The overlapping shaded region (and the two solid lines) represents the solution to the system of inequalities. The tables are:

For \( y\leq x - 4 \):

\( x \)	\( y=x - 4 \)
\( 4 \)	\( 0 \)
\( 5 \)	\( 1 \)

For \( 3x + y\leq4 \):

\( x \)	\( y=-3x + 4 \)
\( 1 \)	\( 1 \)
\( 2 \)	\( -2 \)

(The final graph will have two solid lines and a shaded region of overlap. Since the question asks to graph, the key steps are creating the tables, plotting the points, drawing the lines, and shading the appropriate regions.)