Question
Graph the system of inequalities and determine if the point (3, -3) is a solution.
6
x
+
4
y
<
12
x
>
2
y
≥
−
x
Enter 1 for YES
Enter 2 for NO
(1 point)
a
+
b
=
6
x
+
4
y
<
12
x
>
2
y
≥
−
x
Enter 1 for YES
Enter 2 for NO
(1 point)
a
+
b
=
Answers
GPT-4o mini
Let's analyze the system of inequalities step by step and then determine whether the point \((3, -3)\) is a solution.
1. **First Inequality:**
\[
6x + 4y < 12
\]
To graph this inequality, first convert it to the form of an equation:
\[
6x + 4y = 12
\]
Rearranging gives:
\[
4y = -6x + 12 \quad \Rightarrow \quad y = -\frac{3}{2}x + 3
\]
This line has a y-intercept of 3 and a slope of \(-\frac{3}{2}\). Since the inequality is strict (\(<\)), the area below this line is shaded, and the line itself is dashed.
2. **Second Inequality:**
\[
x > 2
\]
This represents a vertical line at \(x = 2\). The area to the right of this line is shaded, but the line itself is not included (also dashed).
3. **Third Inequality:**
\[
y \geq -x
\]
The corresponding equation is:
\[
y = -x
\]
This line has a y-intercept of 0 and slopes down at an angle of 45 degrees. For this inequality, we shade the area above the line, and this line is included (solid line).
Now let's check if the point \((3, -3)\) satisfies each inequality:
1. **For \(6x + 4y < 12\)**:
\[
6(3) + 4(-3) = 18 - 12 = 6 < 12 \quad \text{(True)}
\]
2. **For \(x > 2\)**:
\[
3 > 2 \quad \text{(True)}
\]
3. **For \(y \geq -x\)**:
\[
-3 \geq -3 \quad \text{(True)}
\]
Since the point \((3, -3)\) satisfies all three inequalities, it is a solution to the system.
Thus, the answer is:
**Enter 1 for YES.**
Finally, you're asking for \(a + b =\), but it seems like specific values for \(a\) and \(b\) are missing. If you have specific numbers for \(a\) and \(b\), you can simply add them, or if you meant something else, please clarify!
1. **First Inequality:**
\[
6x + 4y < 12
\]
To graph this inequality, first convert it to the form of an equation:
\[
6x + 4y = 12
\]
Rearranging gives:
\[
4y = -6x + 12 \quad \Rightarrow \quad y = -\frac{3}{2}x + 3
\]
This line has a y-intercept of 3 and a slope of \(-\frac{3}{2}\). Since the inequality is strict (\(<\)), the area below this line is shaded, and the line itself is dashed.
2. **Second Inequality:**
\[
x > 2
\]
This represents a vertical line at \(x = 2\). The area to the right of this line is shaded, but the line itself is not included (also dashed).
3. **Third Inequality:**
\[
y \geq -x
\]
The corresponding equation is:
\[
y = -x
\]
This line has a y-intercept of 0 and slopes down at an angle of 45 degrees. For this inequality, we shade the area above the line, and this line is included (solid line).
Now let's check if the point \((3, -3)\) satisfies each inequality:
1. **For \(6x + 4y < 12\)**:
\[
6(3) + 4(-3) = 18 - 12 = 6 < 12 \quad \text{(True)}
\]
2. **For \(x > 2\)**:
\[
3 > 2 \quad \text{(True)}
\]
3. **For \(y \geq -x\)**:
\[
-3 \geq -3 \quad \text{(True)}
\]
Since the point \((3, -3)\) satisfies all three inequalities, it is a solution to the system.
Thus, the answer is:
**Enter 1 for YES.**
Finally, you're asking for \(a + b =\), but it seems like specific values for \(a\) and \(b\) are missing. If you have specific numbers for \(a\) and \(b\), you can simply add them, or if you meant something else, please clarify!