To graph the given system of inequalities, we need to visualize the regions that satisfy both inequalities. Let's break down each step to find and graph the solution:
### Step 1: Understanding the Inequalities
1. Inequality 1: [tex]\( y \leq \frac{1}{3}x - 2 \)[/tex]
- This is a linear inequality representing a region below (or on) the line [tex]\( y = \frac{1}{3}x - 2 \)[/tex].
2. Inequality 2: [tex]\( x < 4 \)[/tex]
- This represents the region to the left of the vertical line [tex]\( x = 4 \)[/tex].
### Step 2: Plotting [tex]\( y \leq \frac{1}{3}x - 2 \)[/tex]
1. Rewrite the boundary equation of the inequality: [tex]\( y = \frac{1}{3}x - 2 \)[/tex]
2. Plot the line on a coordinate plane. To do this, find at least two points:
- When [tex]\( x = 0 \)[/tex]:
[tex]\[
y = \frac{1}{3}(0) - 2 = -2 \implies (0, -2)
\][/tex]
- When [tex]\( x = 3 \)[/tex]:
[tex]\[
y = \frac{1}{3}(3) - 2 = 1 - 2 = -1 \implies (3, -1)
\][/tex]
3. Draw a solid line through these points, as the inequality is [tex]\( \leq \)[/tex] (indicating the line itself is included).
4. Shade the region below this line, since the inequality is [tex]\( y \leq \frac{1}{3}x - 2 \)[/tex].
### Step 3: Plotting [tex]\( x < 4 \)[/tex]
1. Draw a vertical dashed line at [tex]\( x=4 \)[/tex]. The dashed line indicates that the boundary [tex]\( x = 4 \)[/tex] is not included in the solution.
2. Shade the region to the left of this line, as the inequality is [tex]\( x < 4 \)[/tex].
### Step 4: Identifying the Solution Region
The solution to the system of inequalities is the region where the shaded areas from both inequalities overlap.
### Step 5: Final Graph
- The boundary line [tex]\( y = \frac{1}{3}x - 2 \)[/tex] is solid.
- The boundary line [tex]\( x = 4 \)[/tex] is dashed.
- The overlapping shaded region represents the solution to the system of inequalities.
Here's a summary of your graph:
1. Draw the vertical dashed line [tex]\( x = 4 \)[/tex].
2. Draw the solid line [tex]\( y = \frac{1}{3}x - 2 \)[/tex].
3. Shade the area below the line [tex]\( y = \frac{1}{3}x - 2 \)[/tex] and to the left of the line [tex]\( x = 4 \)[/tex].
This region represents the points [tex]\((x, y)\)[/tex] that satisfy both inequalities of the system [tex]\( \{ y \leq \frac{1}{3} x - 2, x < 4 \} \)[/tex].
