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Sagot :
To graph the solution to the system of inequalities, let's go through each inequality step-by-step.
### Inequality 1: [tex]\(3y > 2x + 12\)[/tex]
1. Rewrite in slope-intercept form:
[tex]\[y > \frac{2}{3}x + 4\][/tex]
2. Graph the boundary line:
- This inequality represents a region above the line [tex]\(y = \frac{2}{3}x + 4\)[/tex].
- The boundary line [tex]\(y = \frac{2}{3}x + 4\)[/tex] is dashed because the inequality is strict ("greater than" and not "greater than or equal to").
- To draw this line, find two points:
- When [tex]\(x = 0\)[/tex], [tex]\(y = 4\)[/tex]. So, one point is [tex]\((0, 4)\)[/tex].
- When [tex]\(x = 3\)[/tex], [tex]\[ y = \frac{2}{3}(3) + 4 = 2 + 4 = 6 \][/tex]. So, another point is [tex]\((3, 6)\)[/tex].
3. Shade the region:
- Since the inequality is [tex]\(y > \frac{2}{3}x + 4\)[/tex], shade the area above the dashed line.
### Inequality 2: [tex]\(2x + y \leq -5\)[/tex]
1. Rewrite in slope-intercept form:
[tex]\[ y \leq -2x - 5 \][/tex]
2. Graph the boundary line:
- This inequality represents a region below the line [tex]\(y = -2x - 5\)[/tex].
- The boundary line [tex]\(y = -2x - 5\)[/tex] is solid because the inequality includes equality ("less than or equal to").
- To draw this line, find two points:
- When [tex]\(x = 0\)[/tex], [tex]\(y = -5\)[/tex]. So, one point is [tex]\((0, -5)\)[/tex].
- When [tex]\(x = -5\)[/tex], [tex]\[ y = -2(-5) - 5 = 10 - 5 = 5 \][/tex]. So, another point is [tex]\((-5, 5)\)[/tex].
3. Shade the region:
- Since the inequality is [tex]\(y \leq -2x - 5\)[/tex], shade the area below the solid line.
### Solution of the System
- The solution to the system is the region where the shaded areas overlap.
#### Graphing the Solution:
1. Draw your coordinate plane.
2. Plot and draw the line [tex]\(y = \frac{2}{3}x + 4\)[/tex] as a dashed line.
3. Plot and draw the line [tex]\(y = -2x - 5\)[/tex] as a solid line.
4. Shade the region above the dashed line [tex]\(y = \frac{2}{3}x + 4\)[/tex].
5. Shade the region below the solid line [tex]\(y = -2x - 5\)[/tex].
6. The overlapping region is the solution to the system of inequalities.
These steps will give you a clear graphical representation of the solution to the given system of inequalities.
### Inequality 1: [tex]\(3y > 2x + 12\)[/tex]
1. Rewrite in slope-intercept form:
[tex]\[y > \frac{2}{3}x + 4\][/tex]
2. Graph the boundary line:
- This inequality represents a region above the line [tex]\(y = \frac{2}{3}x + 4\)[/tex].
- The boundary line [tex]\(y = \frac{2}{3}x + 4\)[/tex] is dashed because the inequality is strict ("greater than" and not "greater than or equal to").
- To draw this line, find two points:
- When [tex]\(x = 0\)[/tex], [tex]\(y = 4\)[/tex]. So, one point is [tex]\((0, 4)\)[/tex].
- When [tex]\(x = 3\)[/tex], [tex]\[ y = \frac{2}{3}(3) + 4 = 2 + 4 = 6 \][/tex]. So, another point is [tex]\((3, 6)\)[/tex].
3. Shade the region:
- Since the inequality is [tex]\(y > \frac{2}{3}x + 4\)[/tex], shade the area above the dashed line.
### Inequality 2: [tex]\(2x + y \leq -5\)[/tex]
1. Rewrite in slope-intercept form:
[tex]\[ y \leq -2x - 5 \][/tex]
2. Graph the boundary line:
- This inequality represents a region below the line [tex]\(y = -2x - 5\)[/tex].
- The boundary line [tex]\(y = -2x - 5\)[/tex] is solid because the inequality includes equality ("less than or equal to").
- To draw this line, find two points:
- When [tex]\(x = 0\)[/tex], [tex]\(y = -5\)[/tex]. So, one point is [tex]\((0, -5)\)[/tex].
- When [tex]\(x = -5\)[/tex], [tex]\[ y = -2(-5) - 5 = 10 - 5 = 5 \][/tex]. So, another point is [tex]\((-5, 5)\)[/tex].
3. Shade the region:
- Since the inequality is [tex]\(y \leq -2x - 5\)[/tex], shade the area below the solid line.
### Solution of the System
- The solution to the system is the region where the shaded areas overlap.
#### Graphing the Solution:
1. Draw your coordinate plane.
2. Plot and draw the line [tex]\(y = \frac{2}{3}x + 4\)[/tex] as a dashed line.
3. Plot and draw the line [tex]\(y = -2x - 5\)[/tex] as a solid line.
4. Shade the region above the dashed line [tex]\(y = \frac{2}{3}x + 4\)[/tex].
5. Shade the region below the solid line [tex]\(y = -2x - 5\)[/tex].
6. The overlapping region is the solution to the system of inequalities.
These steps will give you a clear graphical representation of the solution to the given system of inequalities.
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