Certainly! Let's graph the given system of inequalities step-by-step:
### Given Inequalities:
1. [tex]\( y \leq \frac{1}{3}x - 2 \)[/tex]
2. [tex]\( x < 4 \)[/tex]
### Step 1: Graph the Boundary Line for the First Inequality
Begin by graphing the line [tex]\( y = \frac{1}{3}x - 2 \)[/tex]. This line serves as the boundary for the inequality [tex]\( y \leq \frac{1}{3}x - 2 \)[/tex].
#### Intercepts:
- Y-intercept: When [tex]\( x = 0 \)[/tex]:
[tex]\[
y = \frac{1}{3}(0) - 2 = -2
\][/tex]
So, the y-intercept is [tex]\( (0, -2) \)[/tex].
- X-intercept: When [tex]\( y = 0 \)[/tex]:
[tex]\[
0 = \frac{1}{3}x - 2 \implies \frac{1}{3}x = 2 \implies x = 6
\][/tex]
So, the x-intercept is [tex]\( (6, 0) \)[/tex].
#### Drawing the Line
Plot the points [tex]\( (0, -2) \)[/tex] and [tex]\( (6, 0) \)[/tex] on a coordinate plane, then draw the line connecting these points. This line has a slope of [tex]\( \frac{1}{3} \)[/tex].
Since the inequality is [tex]\( y \leq \frac{1}{3}x - 2 \)[/tex], shade the region below and on the line.
### Step 2: Graph the Inequality [tex]\( x < 4 \)[/tex]
Draw a vertical dashed line at [tex]\( x = 4 \)[/tex]. A dashed line is used because the inequality is strict (it doesn't include [tex]\( x = 4 \)[/tex]).
Shade to the left of this line because the inequality specifies [tex]\( x < 4 \)[/tex].
### Step 3: Identify the Solution Region
The solution to this system of inequalities is the region where the shaded areas overlap:
- Below and on the line [tex]\( y = \frac{1}{3}x - 2 \)[/tex]
- To the left of [tex]\( x = 4 \)[/tex]
### Graph Summary
1. Plot the line [tex]\( y = \frac{1}{3}x - 2 \)[/tex]. Shade below this line (including the line since it is [tex]\( \leq \)[/tex]).
2. Draw a dashed vertical line at [tex]\( x = 4 \)[/tex]. Shade to the left of this line (not including the line itself).
3. The overlapping shaded region of these two inequalities is the solution to the system.
### Graph:
1. Boundary: Line [tex]\( y = \frac{1}{3}x - 2 \)[/tex]
- Y-intercept at [tex]\( (0, -2) \)[/tex]
- X-intercept at [tex]\( (6, 0) \)[/tex]
2. Vertical Boundary: Dashed line at [tex]\( x = 4 \)[/tex]
3. Shading:
- Below the line [tex]\( y = \frac{1}{3}x - 2 \)[/tex]
- To the left of [tex]\( x = 4 \)[/tex]
This overlapping region shows all possible [tex]\( (x, y) \)[/tex] pairs that satisfy both inequalities.
