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  3. On the number line shown above, the tick marks are equally spaced. Which of the following statements about the numbers x, y, and

On the number line shown above, the tick marks are equally spaced. Which of the following statements about the numbers x, y, and

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问题
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.
选项 A、xyz<0
B、x + z =y
C、z(y-x)>0
答案A，B，C
解析 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.
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