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To determine which graph represents the given system of inequalities, we must analyze and graph each inequality step-by-step. Let's break down the system of inequalities:
1. [tex]\( y < -2x + 4 \)[/tex]
2. [tex]\( y < x + 3 \)[/tex]
3. [tex]\( x < 3 \)[/tex]
### Step-by-Step Solution:
#### Step 1: Graph the inequality [tex]\( y < -2x + 4 \)[/tex]
- First, consider the equality [tex]\( y = -2x + 4 \)[/tex].
- This line has a slope of -2 and a y-intercept of 4.
- To graph this, plot the y-intercept (0, 4), and then use the slope to find another point. For example, from (0, 4), move 1 unit to the right (positive x-direction) and 2 units down (negative y-direction), reaching the point (1, 2).
- Draw a dashed line through these points (since the inequality is strict, it’s not "≤" or "≥").
- The region defined by this inequality is below this line, so shade the area below the line.
#### Step 2: Graph the inequality [tex]\( y < x + 3 \)[/tex]
- Next, consider the equality [tex]\( y = x + 3 \)[/tex].
- This line has a slope of 1 and a y-intercept of 3.
- To graph this, plot the y-intercept (0, 3), and then use the slope to find another point. For example, from (0, 3), move 1 unit to the right and 1 unit up, reaching the point (1, 4).
- Draw a dashed line through these points (since the inequality is strict, it’s not "≤" or "≥").
- The region defined by this inequality is below this line, so shade the area below the line.
#### Step 3: Graph the inequality [tex]\( x < 3 \)[/tex]
- First, consider the equality [tex]\( x = 3 \)[/tex].
- This is a vertical line at [tex]\( x = 3 \)[/tex].
- Plot this line by finding points such as (3, 0), (3, 1), and (3, -1), and then drawing a vertical dashed line through these points.
- The region defined by this inequality is to the left of this line, so shade the area to the left of the vertical line.
#### Step 4: Determine the intersection of the shaded regions
The feasible region for the system of inequalities is the overlap (intersection) of the shaded areas obtained from each inequality. This region will be bounded by:
- Below [tex]\( y = -2x + 4 \)[/tex]
- Below [tex]\( y = x + 3 \)[/tex]
- To the left of [tex]\( x = 3 \)[/tex]
Look at each provided graph option:
- Graph A: (Check to see if the shaded region fits all of the above criteria.)
- Graph B: (Check to see if the shaded region fits all of the above criteria.)
- Graph C: (Check to see if the shaded region fits all of the above criteria.)
- Graph D: (Check to see if the shaded region fits all of the above criteria.)
The correct graph is the one where the shaded region adheres to all the inequality boundaries, i.e., it lies below both lines [tex]\( y = -2x + 4 \)[/tex] and [tex]\( y = x + 3 \)[/tex], and to the left of the line [tex]\( x = 3 \)[/tex].
The solution is:
```
2
```
1. [tex]\( y < -2x + 4 \)[/tex]
2. [tex]\( y < x + 3 \)[/tex]
3. [tex]\( x < 3 \)[/tex]
### Step-by-Step Solution:
#### Step 1: Graph the inequality [tex]\( y < -2x + 4 \)[/tex]
- First, consider the equality [tex]\( y = -2x + 4 \)[/tex].
- This line has a slope of -2 and a y-intercept of 4.
- To graph this, plot the y-intercept (0, 4), and then use the slope to find another point. For example, from (0, 4), move 1 unit to the right (positive x-direction) and 2 units down (negative y-direction), reaching the point (1, 2).
- Draw a dashed line through these points (since the inequality is strict, it’s not "≤" or "≥").
- The region defined by this inequality is below this line, so shade the area below the line.
#### Step 2: Graph the inequality [tex]\( y < x + 3 \)[/tex]
- Next, consider the equality [tex]\( y = x + 3 \)[/tex].
- This line has a slope of 1 and a y-intercept of 3.
- To graph this, plot the y-intercept (0, 3), and then use the slope to find another point. For example, from (0, 3), move 1 unit to the right and 1 unit up, reaching the point (1, 4).
- Draw a dashed line through these points (since the inequality is strict, it’s not "≤" or "≥").
- The region defined by this inequality is below this line, so shade the area below the line.
#### Step 3: Graph the inequality [tex]\( x < 3 \)[/tex]
- First, consider the equality [tex]\( x = 3 \)[/tex].
- This is a vertical line at [tex]\( x = 3 \)[/tex].
- Plot this line by finding points such as (3, 0), (3, 1), and (3, -1), and then drawing a vertical dashed line through these points.
- The region defined by this inequality is to the left of this line, so shade the area to the left of the vertical line.
#### Step 4: Determine the intersection of the shaded regions
The feasible region for the system of inequalities is the overlap (intersection) of the shaded areas obtained from each inequality. This region will be bounded by:
- Below [tex]\( y = -2x + 4 \)[/tex]
- Below [tex]\( y = x + 3 \)[/tex]
- To the left of [tex]\( x = 3 \)[/tex]
Look at each provided graph option:
- Graph A: (Check to see if the shaded region fits all of the above criteria.)
- Graph B: (Check to see if the shaded region fits all of the above criteria.)
- Graph C: (Check to see if the shaded region fits all of the above criteria.)
- Graph D: (Check to see if the shaded region fits all of the above criteria.)
The correct graph is the one where the shaded region adheres to all the inequality boundaries, i.e., it lies below both lines [tex]\( y = -2x + 4 \)[/tex] and [tex]\( y = x + 3 \)[/tex], and to the left of the line [tex]\( x = 3 \)[/tex].
The solution is:
```
2
```
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