Question
A system of linear inequalities is given.
{1/2x+6>2y
{1/3y<2/3x
Which graph shows the solution to this system of inequalities?
A.
A system of linear inequalities with two solid lines is shown. The first solid line passes through the points (negative 4, negative 8), (2, 4), and (4, 8). The second solid line passes through the points (negative 8, 1), (4, 4) and (8, 5). A shaded common region is shown below the lines.
B.
A system of linear inequalities with two dashed lines is shown. The first dashed line passes through the points (negative 4, negative 8), (2, 4), and (4, 8). The second dashed line passes through the points (negative 8, 1), (4, 4) and (8, 5). A shaded common region is shown below the lines.
C.
A system of linear inequalities with two solid lines is shown. The first solid line passes through the points (negative 4, negative 8), (2, 4), and (4, 8). The second solid line passes through the points (negative 8, 1), (4, 4) and (8, 5). A shaded common region is shown above the lines.
D.
A system of linear inequalities with two dashed lines is shown. The first dashed line passes through the points (negative 4, negative 8), (2, 4), and (4, 8). The second dashed line passes through the points (negative 8, 1), (4, 4) and (8, 5). A shaded common region is shown above the lines.
{1/2x+6>2y
{1/3y<2/3x
Which graph shows the solution to this system of inequalities?
A.
A system of linear inequalities with two solid lines is shown. The first solid line passes through the points (negative 4, negative 8), (2, 4), and (4, 8). The second solid line passes through the points (negative 8, 1), (4, 4) and (8, 5). A shaded common region is shown below the lines.
B.
A system of linear inequalities with two dashed lines is shown. The first dashed line passes through the points (negative 4, negative 8), (2, 4), and (4, 8). The second dashed line passes through the points (negative 8, 1), (4, 4) and (8, 5). A shaded common region is shown below the lines.
C.
A system of linear inequalities with two solid lines is shown. The first solid line passes through the points (negative 4, negative 8), (2, 4), and (4, 8). The second solid line passes through the points (negative 8, 1), (4, 4) and (8, 5). A shaded common region is shown above the lines.
D.
A system of linear inequalities with two dashed lines is shown. The first dashed line passes through the points (negative 4, negative 8), (2, 4), and (4, 8). The second dashed line passes through the points (negative 8, 1), (4, 4) and (8, 5). A shaded common region is shown above the lines.
Answers
GPT-4o mini
To determine which graph represents the solution to the given system of linear inequalities, we need to analyze the inequalities one by one.
1. **First Inequality:**
\(\frac{1}{2}x + 6 > 2y\)
Rearranging gives:
\(2y < \frac{1}{2}x + 6\)
or
\(y < \frac{1}{4}x + 3\)
This indicates that the region of solutions is below the line \(y = \frac{1}{4}x + 3\). The line itself is dashed because the original inequality is greater than (not equal to).
2. **Second Inequality:**
\(\frac{1}{3}y < \frac{2}{3}x\)
Rearranging gives:
\(y < 2x\)
This indicates that the region of solutions is below the line \(y = 2x\). Like the first one, this line is also dashed because it is a strict inequality.
Now, let's summarize these findings:
- Both lines in the graphs should be dashed lines.
- The solution region is below both lines.
Considering the options:
- **Option A** has solid lines and shaded below the lines—incorrect.
- **Option B** has dashed lines and shaded below the lines—correct.
- **Option C** has solid lines and shaded above the lines—incorrect.
- **Option D** has dashed lines but shaded above the lines—incorrect.
Thus, the correct answer is **B.**
1. **First Inequality:**
\(\frac{1}{2}x + 6 > 2y\)
Rearranging gives:
\(2y < \frac{1}{2}x + 6\)
or
\(y < \frac{1}{4}x + 3\)
This indicates that the region of solutions is below the line \(y = \frac{1}{4}x + 3\). The line itself is dashed because the original inequality is greater than (not equal to).
2. **Second Inequality:**
\(\frac{1}{3}y < \frac{2}{3}x\)
Rearranging gives:
\(y < 2x\)
This indicates that the region of solutions is below the line \(y = 2x\). Like the first one, this line is also dashed because it is a strict inequality.
Now, let's summarize these findings:
- Both lines in the graphs should be dashed lines.
- The solution region is below both lines.
Considering the options:
- **Option A** has solid lines and shaded below the lines—incorrect.
- **Option B** has dashed lines and shaded below the lines—correct.
- **Option C** has solid lines and shaded above the lines—incorrect.
- **Option D** has dashed lines but shaded above the lines—incorrect.
Thus, the correct answer is **B.**