Answer: Graph D
Open holes at 1 and 3; shading in between
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Explanation:
Let p = x-1 and q = x-3
The problem we want to solve is in the form p/q < 0
Since p/q is negative, there are two cases we have to address
- Case 1) p is positive and q is negative, or,
- Case 2) p is negative and q is positive
The signs of p and q are opposite one another. Recall this is due to the fact that dividing a positive over a negative (or vice versa) leads to a negative value.
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Let's go over Case 1
If p is positive, then
p > 0
x-1 > 0
x > 1
If q is negative, then
q < 0
x-3 < 0
x < 3
Combine x > 1 with x < 3 and we get 1 < x < 3. We'll come back to this later.
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Now case 2
If p is negative, then
p < 0
x-1 < 0
x < 1
If q is positive, then,
q > 0
x-3 > 0
x > 3
We get a contradiction.
Both "p is negative" AND "q is positive" must be true at the same time. But if p is negative, then x < 1 which contradicts x > 3 (which happens when q is positive)
Put simply: case 2 is not possible. There's no way to have p be negative at the same time q is positive. This because we're wanting x to both be smaller than 1 AND also be greater than 3 at the same time.
We must rule out case 2 and side with case 1 instead.
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The conclusion of case 1 was that 1 < x < 3
The graph of this has open holes at 1 and 3 on the number line. The shading is between the open holes to represent all values between 1 and 3, excluding the endpoints. Graph D indicates these features, so that's why graph D is the answer.
Note: the open holes indicate that x = 1 and x = 3 are not part of the solution set. Another reason x = 3 is restricted is to avoid a division by zero error.
Answer:
- It is the last graph: solid line, shaded area over the line x = 2 - x/2
Explanation:
1) <u>Set the algebraigic expression that represents the combinations of sofa and pillow orders:</u>
- Number of sofas: x (given)
- Number of pillows: 2y (given, since they come in pairs)
- Number of items = number of sofas + number of pillows = x + 2y
- Minimum of 4 items in each order (given) ⇒ x + 2y ≥ 4
<u>2) Predict the graph of the inequality x + 2y ≥ 4</u>
- The border line is the equation x + 2y = 4
- You can choose two points to draw a line
- Choose the axis-intercepst:
x = 0 ⇒ 2y = 4 ⇒ y =4/2 ⇒ y = 2 ⇒ point (0,2)
y = 0 ⇒ x = 4 ⇒ point (4,0)
Then the lines goes through (0,2) and (4,0) ... [the four graphs meet this]
- The shading area is above the line because when you solve for y you get y ≥ 2 - x/2, and the line is included because the "equal to" part of the symbol (≥ means greater than or equal to).
- To state that the line is included the graph uses a continous line instead of a dotted one.
<u>3) Conclusion:</u>
That means that the correct graph is the last one: solid line, shaded area over the line y = 2 - x/2.
Note: a more detailed graph would include the fact that the items cannot be negative, i.e. x ≥ 0 and y ≥ 0, which would result in that the shaded area would be on the first quadrant.
Answer:
25 4-point questions and 75 2-point questions
Step-by-step explanation:
Let x = no. of 4-point questions
Then 3x = no. of 2-point questions
So, x + 3x = 100
4x = 100
x = 25 = no. of 4-point questions
and 3x = 3(25) = 75 = no. of 2-point questions
Answer:
A.
Step-by-step explanation:
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