Answer :
Sure, let's graph the solution to this system of inequalities step-by-step.
### Step 1: Graph the line [tex]\( y = 2x + 4 \)[/tex]
1. Identify the y-intercept and slope:
- The y-intercept is [tex]\( 4 \)[/tex] (point [tex]\((0, 4)\)[/tex]).
- The slope is [tex]\( 2 \)[/tex], which means the line rises by 2 units for every 1 unit it moves to the right.
2. Plot another point using the slope:
- From the y-intercept [tex]\((0, 4)\)[/tex], move right by 1 unit and up by 2 units to the point [tex]\((1, 6)\)[/tex].
3. Draw the line through these two points. Use a dashed line because the inequality is strict ([tex]\( y > 2x + 4 \)[/tex]).
4. Shade the region above the dashed line because the inequality is [tex]\( y > 2x + 4 \)[/tex].
### Step 2: Graph the line [tex]\( x + y = 6 \)[/tex]
1. Rewrite in slope-intercept form: [tex]\( y = -x + 6 \)[/tex].
2. Identify the y-intercept and slope:
- The y-intercept is [tex]\( 6 \)[/tex] (point [tex]\((0, 6)\)[/tex]).
- The slope is [tex]\( -1 \)[/tex], which means the line falls by 1 unit for every 1 unit it moves to the right.
3. Plot another point using the slope:
- From the y-intercept [tex]\((0, 6)\)[/tex], move right by 1 unit and down by 1 unit to the point [tex]\((1, 5)\)[/tex].
4. Draw the line through these two points. Use a solid line because the inequality includes equality ([tex]\( x + y \leq 6 \)[/tex]).
5. Shade the region below the line because the inequality is [tex]\( x + y \leq 6 \)[/tex].
### Step 3: Find the Overlapping Region
1. The solution to the system of inequalities is where the shaded regions from steps 1 and 2 overlap.
2. Identify the overlapping region:
- Above the line [tex]\( y = 2x + 4 \)[/tex].
- Below the line [tex]\( x + y = 6 \)[/tex].
3. This overlapping area represents the solution to the system of inequalities.
### Coordinate Plane Summary
1. Dashed Line: [tex]\( y = 2x + 4 \)[/tex]. Shade above this line.
2. Solid Line: [tex]\( x + y = 6 \)[/tex]. Shade below this line.
3. Solution Region: The overlapping area above [tex]\( y = 2x + 4 \)[/tex] and below [tex]\( x + y = 6 \)[/tex].
### Visual Summary
```
|
20 -|
|
|
|
15 -|
|
|
|
10 -| . . . . . . . . . . . .
| /
| /
5 -| - - - - - - - (0,4) . . . . . . . . . . . .
| / / |
| / / |
0 -| . * . . . . . . . . . . . . . . . . . . . . . . . .
| / \ (1,5) / /
| / \ / /
-5 -|-----|-----------|------------------|-------(1,0)|(6,0)
|
|
-10-|
|============================================================
-10 -5 0 5 10
```
The shaded region above the dashed line [tex]\( y = 2x + 4 \)[/tex] and below the solid line [tex]\( x + y = 6 \)[/tex] is our solution.
### Step 1: Graph the line [tex]\( y = 2x + 4 \)[/tex]
1. Identify the y-intercept and slope:
- The y-intercept is [tex]\( 4 \)[/tex] (point [tex]\((0, 4)\)[/tex]).
- The slope is [tex]\( 2 \)[/tex], which means the line rises by 2 units for every 1 unit it moves to the right.
2. Plot another point using the slope:
- From the y-intercept [tex]\((0, 4)\)[/tex], move right by 1 unit and up by 2 units to the point [tex]\((1, 6)\)[/tex].
3. Draw the line through these two points. Use a dashed line because the inequality is strict ([tex]\( y > 2x + 4 \)[/tex]).
4. Shade the region above the dashed line because the inequality is [tex]\( y > 2x + 4 \)[/tex].
### Step 2: Graph the line [tex]\( x + y = 6 \)[/tex]
1. Rewrite in slope-intercept form: [tex]\( y = -x + 6 \)[/tex].
2. Identify the y-intercept and slope:
- The y-intercept is [tex]\( 6 \)[/tex] (point [tex]\((0, 6)\)[/tex]).
- The slope is [tex]\( -1 \)[/tex], which means the line falls by 1 unit for every 1 unit it moves to the right.
3. Plot another point using the slope:
- From the y-intercept [tex]\((0, 6)\)[/tex], move right by 1 unit and down by 1 unit to the point [tex]\((1, 5)\)[/tex].
4. Draw the line through these two points. Use a solid line because the inequality includes equality ([tex]\( x + y \leq 6 \)[/tex]).
5. Shade the region below the line because the inequality is [tex]\( x + y \leq 6 \)[/tex].
### Step 3: Find the Overlapping Region
1. The solution to the system of inequalities is where the shaded regions from steps 1 and 2 overlap.
2. Identify the overlapping region:
- Above the line [tex]\( y = 2x + 4 \)[/tex].
- Below the line [tex]\( x + y = 6 \)[/tex].
3. This overlapping area represents the solution to the system of inequalities.
### Coordinate Plane Summary
1. Dashed Line: [tex]\( y = 2x + 4 \)[/tex]. Shade above this line.
2. Solid Line: [tex]\( x + y = 6 \)[/tex]. Shade below this line.
3. Solution Region: The overlapping area above [tex]\( y = 2x + 4 \)[/tex] and below [tex]\( x + y = 6 \)[/tex].
### Visual Summary
```
|
20 -|
|
|
|
15 -|
|
|
|
10 -| . . . . . . . . . . . .
| /
| /
5 -| - - - - - - - (0,4) . . . . . . . . . . . .
| / / |
| / / |
0 -| . * . . . . . . . . . . . . . . . . . . . . . . . .
| / \ (1,5) / /
| / \ / /
-5 -|-----|-----------|------------------|-------(1,0)|(6,0)
|
|
-10-|
|============================================================
-10 -5 0 5 10
```
The shaded region above the dashed line [tex]\( y = 2x + 4 \)[/tex] and below the solid line [tex]\( x + y = 6 \)[/tex] is our solution.
