Find the coordinates of the vertices of the figures formed by y -< x + 2, x + 2 -< 6, and y >- -2
A)(0,0),(2,4),(8,-2)
B)(-4,-2),(2,4),(8,-2)
C)(-4,-2),(4,2),(8,-2)
D)(-2,-4),(2,4),(8,-2)
I chose B
this one confused me this is all the work to show:
-2 -< -4 + 2
2 + 4 -< 6
-2
the way u did it, if I try to do the others, Aren't they all true though?
No, all are not true
We have this:
(y -< x + 2) = point 1.
(x + 2 -< 6) = point 2.
(y >- -2) = point 3.
Replace all inequality symbols with an equal sign for easy simplification.
Point 3:
y >- -2 becomes y = -2
We just found y.
We plug y = -2 in point 1 to find x.
-2 = x + 2
-2 - 2 = x
-4 = x
We know that y = -2 and x = -4.
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Notice that the next point to follow in choice B is (2,4), where x = 2 and y = 4.
We plug that into all three points to see if we get a true statement.
(y -< x + 2) = point 1.
4 -< 2 + 2...true statement
(x + 2 -< 6) = point 2.
2 + 4 -< 6...true statement
(y >- -2) = point 3.
4 >- -2...true statement
So far, it appears that choice B is correct.
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What is the next point given in choice B? It is point (8, -2).
We do the same-->plug to check for true statement.
(y -< x + 2) = point 1.
-2 -< 8 + 2...true statement
(x + 2 -< 6) = point 2.
8 + 2 -< 6...false statement
(y >- -2) = point 3.
-2 >- -2...true statement
We know that choice A and D cannot be the answers. We just found out that choice B is most likely not the answer because of the false statement.
I would say the answer is choice C.
To find the coordinates of the vertices of the figures formed by the given inequalities, follow these steps:
Step 1: Plot the equations on a graph.
The inequalities given are:
y ≤ x + 2 (Equation 1)
x + 2 ≤ 6 (Equation 2)
y > -2 (Equation 3)
To plot these inequalities, we draw the corresponding lines on a graph:
- For Equation 1: y = x + 2, which has a slope of 1 and a y-intercept of 2.
- For Equation 2: x = 4, which is a vertical line passing through x = 4.
- For Equation 3: y > -2, which is a horizontal line parallel to the x-axis passing above y = -2.
Step 2: Determine the vertices.
The vertices of the figure are the points where the lines intersect or where two lines meet.
To find the vertices, we need to find the intersection points of the lines.
From the graph, you can see that the lines intersect at two points:
- Point A: (2, 4)
- Point B: (8, -2)
Step 3: Verify all vertices.
Now, we need to check if these points are valid solutions for the given inequalities.
Substitute the coordinates of Point A and Point B into the inequalities:
- For Point A: (2, 4)
- Equation 1: 4 ≤ 2 + 2 [True]
- Equation 2: 2 + 2 ≤ 6 [True]
- Equation 3: 4 > -2 [True]
- For Point B: (8, -2)
- Equation 1: -2 ≤ 8 + 2 [True]
- Equation 2: 8 + 2 ≤ 6 [False]
- Equation 3: -2 > -2 [False]
Based on the above calculations, we can conclude that Point A (2, 4) is a valid vertex, but Point B (8, -2) is not.
Therefore, the correct answer is option B: (-4, -2), (2, 4), (8, -2).