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18 May 2024, 10:54
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Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
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0% (00:00) correct
100% (00:56) wrong based on 2 sessions
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On the number line shown above, the tick marks are equally spaced. Which of the following statements about the numbers x, y, and z must be true?
Indicate all such statements.
A. \(xyz<0\)
B. \(x + z = y\)
C. \(z(y-x)>0\)
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20 May 2024, 01:51
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Concept:
(negative)(negative) = positive
(positive)(negative) = negative
(negative)(negative) = positive
No exceptions!
Thus:
1) (-)(+)(+) = + > 0
2) plugging in numbers is easier for this one.
3) (+)(positive - negative) = (+)(+--) = (+)(+) = + > 0
(negative)(negative) = positive
(positive)(negative) = negative
(negative)(negative) = positive
No exceptions!
Thus:
1) (-)(+)(+) = + > 0
2) plugging in numbers is easier for this one.
3) (+)(positive - negative) = (+)(+--) = (+)(+) = + > 0
25 Jun 2024, 09:29
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Official Explanation
You can see from their positions on the number line that x is less than 0 and both y and z are greater than 0. Because the tick marks are equally spaced, you can also see that x = -y and z = 2y. You need to evaluate each answer choice separately to determine whether it must be true.
Choice A says that the product of the three numbers x, y, and z is less than 0. Recall that the product of three numbers is negative under either of the following two conditions.
Choice A must be true, since x is negative and y and z are positive. Choice B is the equation x + z = y. To see whether the equation must be true, it is a good idea to express two of the variables in terms of the third (that is, to “get rid of” two of the variables). The equations x = -y and z = 2y give x and z in terms of y, so the equation x + z = y can be rewritten, substituting -y for x and 2y for z, as -y + 2y = y. In this form you can quickly conclude that the equation must be true.
Choice C says that the product of the two numbers z and y - x is greater than 0. Recall that the product of two numbers is positive under either of the following two conditions.
Since you already know that z is positive, you can conclude that the product z(y - x) will be positive if y - x is positive. By adding x to both sides of the inequality y-x>0, you can see that it is equivalent to the inequality y>x, which is clearly true from the number line. Since y - x is positive, the product z(y - x) must be positive.
Therefore, the correct answer consists of Choices A, B, and C.
You can see from their positions on the number line that x is less than 0 and both y and z are greater than 0. Because the tick marks are equally spaced, you can also see that x = -y and z = 2y. You need to evaluate each answer choice separately to determine whether it must be true.
Choice A says that the product of the three numbers x, y, and z is less than 0. Recall that the product of three numbers is negative under either of the following two conditions.
- All three numbers are negative.
- One of the numbers is negative and the other two numbers are positive.
Choice A must be true, since x is negative and y and z are positive. Choice B is the equation x + z = y. To see whether the equation must be true, it is a good idea to express two of the variables in terms of the third (that is, to “get rid of” two of the variables). The equations x = -y and z = 2y give x and z in terms of y, so the equation x + z = y can be rewritten, substituting -y for x and 2y for z, as -y + 2y = y. In this form you can quickly conclude that the equation must be true.
Choice C says that the product of the two numbers z and y - x is greater than 0. Recall that the product of two numbers is positive under either of the following two conditions.
- Both numbers are positive.
- Both numbers are negative.
Since you already know that z is positive, you can conclude that the product z(y - x) will be positive if y - x is positive. By adding x to both sides of the inequality y-x>0, you can see that it is equivalent to the inequality y>x, which is clearly true from the number line. Since y - x is positive, the product z(y - x) must be positive.
Therefore, the correct answer consists of Choices A, B, and C.
