Answer :
To determine which graph correctly represents the system of inequalities:
1. Understand the individual inequalities:
- The first inequality is [tex]\( y < -\frac{1}{3}x + 1 \)[/tex].
- The second inequality is [tex]\( y \leq 2x - 3 \)[/tex].
2. Graph each inequality separately before combining them:
### Graphing [tex]\( y < -\frac{1}{3}x + 1 \)[/tex]:
- This inequality represents a line with a slope of [tex]\(-\frac{1}{3}\)[/tex] and a y-intercept of [tex]\(1\)[/tex].
- To graph [tex]\( y < -\frac{1}{3}x + 1 \)[/tex]:
1. Start from the y-intercept at (0, 1).
2. From this point, move down 1 unit and right 3 units to follow the slope of [tex]\(-\frac{1}{3}\)[/tex].
3. Draw a dashed line through these points to indicate that points on the line itself are not included in the solution (since it's a strict inequality "<").
4. Shade the region below this line because [tex]\( y \)[/tex] is less than the line.
### Graphing [tex]\( y \leq 2x - 3 \)[/tex]:
- This inequality represents a line with a slope of [tex]\(2\)[/tex] and a y-intercept of [tex]\(-3\)[/tex].
- To graph [tex]\( y \leq 2x - 3 \)[/tex]:
1. Start from the y-intercept at (0, -3).
2. From this point, move up 2 units and right 1 unit to follow the slope of [tex]\(2\)[/tex].
3. Draw a solid line through these points to indicate that points on the line itself are included in the solution (since it's a non-strict inequality "≤").
4. Shade the region below this line because [tex]\( y \)[/tex] is less than or equal to the line.
### Finding the Solution Region:
- The solution to the system of inequalities is the overlap of the shaded regions from the two graphs.
- This overlap region is where both inequalities are true simultaneously.
### Identifying the Correct Graph:
- Look at the intersection of the shaded areas. You should see:
- A dashed line for [tex]\( y < -\frac{1}{3}x + 1 \)[/tex] and shading below this line.
- A solid line for [tex]\( y \leq 2x - 3 \)[/tex] and shading below this line.
- The overlap will be the region where both shaded areas intersect.
By comparing these conditions to the graphs provided, select the graph where:
1. There is a dashed line starting from (0, 1) with a slope of [tex]\(-\frac{1}{3}\)[/tex] and shading below it.
2. There is a solid line starting from (0, -3) with a slope of [tex]\(2\)[/tex] and shading below it.
3. The shaded region below [tex]\( y < -\frac{1}{3}x + 1 \)[/tex] intersects with the shaded region below [tex]\( y \leq 2x - 3 \)[/tex].
The graph meeting these criteria represents the solution to the system of inequalities.
1. Understand the individual inequalities:
- The first inequality is [tex]\( y < -\frac{1}{3}x + 1 \)[/tex].
- The second inequality is [tex]\( y \leq 2x - 3 \)[/tex].
2. Graph each inequality separately before combining them:
### Graphing [tex]\( y < -\frac{1}{3}x + 1 \)[/tex]:
- This inequality represents a line with a slope of [tex]\(-\frac{1}{3}\)[/tex] and a y-intercept of [tex]\(1\)[/tex].
- To graph [tex]\( y < -\frac{1}{3}x + 1 \)[/tex]:
1. Start from the y-intercept at (0, 1).
2. From this point, move down 1 unit and right 3 units to follow the slope of [tex]\(-\frac{1}{3}\)[/tex].
3. Draw a dashed line through these points to indicate that points on the line itself are not included in the solution (since it's a strict inequality "<").
4. Shade the region below this line because [tex]\( y \)[/tex] is less than the line.
### Graphing [tex]\( y \leq 2x - 3 \)[/tex]:
- This inequality represents a line with a slope of [tex]\(2\)[/tex] and a y-intercept of [tex]\(-3\)[/tex].
- To graph [tex]\( y \leq 2x - 3 \)[/tex]:
1. Start from the y-intercept at (0, -3).
2. From this point, move up 2 units and right 1 unit to follow the slope of [tex]\(2\)[/tex].
3. Draw a solid line through these points to indicate that points on the line itself are included in the solution (since it's a non-strict inequality "≤").
4. Shade the region below this line because [tex]\( y \)[/tex] is less than or equal to the line.
### Finding the Solution Region:
- The solution to the system of inequalities is the overlap of the shaded regions from the two graphs.
- This overlap region is where both inequalities are true simultaneously.
### Identifying the Correct Graph:
- Look at the intersection of the shaded areas. You should see:
- A dashed line for [tex]\( y < -\frac{1}{3}x + 1 \)[/tex] and shading below this line.
- A solid line for [tex]\( y \leq 2x - 3 \)[/tex] and shading below this line.
- The overlap will be the region where both shaded areas intersect.
By comparing these conditions to the graphs provided, select the graph where:
1. There is a dashed line starting from (0, 1) with a slope of [tex]\(-\frac{1}{3}\)[/tex] and shading below it.
2. There is a solid line starting from (0, -3) with a slope of [tex]\(2\)[/tex] and shading below it.
3. The shaded region below [tex]\( y < -\frac{1}{3}x + 1 \)[/tex] intersects with the shaded region below [tex]\( y \leq 2x - 3 \)[/tex].
The graph meeting these criteria represents the solution to the system of inequalities.
