Answer :
To determine which graph represents the system of inequalities, we need to carefully break down the details step-by-step.
### Step 1: Interpret the Inequalities
1. First Inequality: [tex]\( y \geq -5x + 2 \)[/tex]
- This represents a line with a slope of -5 and a y-intercept of 2.
- The "≥" symbol indicates that the region of interest lies above this line, including the line itself.
2. Second Inequality: [tex]\( y > 3x - 1.5 \)[/tex]
- This represents a line with a slope of 3 and a y-intercept of -1.5.
- The ">" symbol indicates that the region of interest lies above this line, but not including the line itself.
### Step 2: Find the Point of Intersection
Next, we calculate the intersection point of these two lines, as this will help us understand the feasible region. The calculated point of intersection from solving the system yields:
[tex]\[ x = 0.4375 \][/tex]
[tex]\[ y = -0.1875 \][/tex]
### Step 3: Plot the Lines
1. Line [tex]\( y = -5x + 2 \)[/tex]:
- Plot the y-intercept (0, 2).
- Use the slope -5 to find another point: for example, from (0, 2) if we move 1 unit right on the x-axis (to x = 1), we move 5 units down (since the slope is negative 5), which gives us point (1, -3).
2. Line [tex]\( y = 3x - 1.5 \)[/tex]:
- Plot the y-intercept (0, -1.5).
- Use the slope 3 to find another point: for example, from (0, -1.5) if we move 1 unit right on the x-axis (to x = 1), we move 3 units up, which gives us point (1, 1.5).
### Step 4: Shade the Regions
1. For [tex]\( y \geq -5x + 2 \)[/tex]:
- Shade the region above this line, including the line itself.
2. For [tex]\( y > 3x - 1.5 \)[/tex]:
- Shade the region above this line, but not including the line itself.
### Step 5: Identify the Feasible Region
Finally, the feasible region is the area where the shaded regions from both inequalities overlap. The points in this region satisfy both inequalities.
### Conclusion
The correct graph representing the system should:
- Show two lines intersecting at approximately (0.4375, -0.1875).
- Have the region above [tex]\( y \geq -5x + 2 \)[/tex] shaded (including the line itself).
- Have the region above [tex]\( y > 3x - 1.5 \)[/tex] shaded (excluding the line itself).
- The intersection area formed by both shaded regions will represent the solution set for the system of inequalities.
This visual technique will help in identifying the correct graph. Look for the graph where these conditions are satisfied.
### Step 1: Interpret the Inequalities
1. First Inequality: [tex]\( y \geq -5x + 2 \)[/tex]
- This represents a line with a slope of -5 and a y-intercept of 2.
- The "≥" symbol indicates that the region of interest lies above this line, including the line itself.
2. Second Inequality: [tex]\( y > 3x - 1.5 \)[/tex]
- This represents a line with a slope of 3 and a y-intercept of -1.5.
- The ">" symbol indicates that the region of interest lies above this line, but not including the line itself.
### Step 2: Find the Point of Intersection
Next, we calculate the intersection point of these two lines, as this will help us understand the feasible region. The calculated point of intersection from solving the system yields:
[tex]\[ x = 0.4375 \][/tex]
[tex]\[ y = -0.1875 \][/tex]
### Step 3: Plot the Lines
1. Line [tex]\( y = -5x + 2 \)[/tex]:
- Plot the y-intercept (0, 2).
- Use the slope -5 to find another point: for example, from (0, 2) if we move 1 unit right on the x-axis (to x = 1), we move 5 units down (since the slope is negative 5), which gives us point (1, -3).
2. Line [tex]\( y = 3x - 1.5 \)[/tex]:
- Plot the y-intercept (0, -1.5).
- Use the slope 3 to find another point: for example, from (0, -1.5) if we move 1 unit right on the x-axis (to x = 1), we move 3 units up, which gives us point (1, 1.5).
### Step 4: Shade the Regions
1. For [tex]\( y \geq -5x + 2 \)[/tex]:
- Shade the region above this line, including the line itself.
2. For [tex]\( y > 3x - 1.5 \)[/tex]:
- Shade the region above this line, but not including the line itself.
### Step 5: Identify the Feasible Region
Finally, the feasible region is the area where the shaded regions from both inequalities overlap. The points in this region satisfy both inequalities.
### Conclusion
The correct graph representing the system should:
- Show two lines intersecting at approximately (0.4375, -0.1875).
- Have the region above [tex]\( y \geq -5x + 2 \)[/tex] shaded (including the line itself).
- Have the region above [tex]\( y > 3x - 1.5 \)[/tex] shaded (excluding the line itself).
- The intersection area formed by both shaded regions will represent the solution set for the system of inequalities.
This visual technique will help in identifying the correct graph. Look for the graph where these conditions are satisfied.
