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atlantic · Sun Jun 29, 2008 10:18 am

Hi guys,

Your help is kindly requested for this one,

post OA latter

Ian Stewart · Sun Jun 29, 2008 1:03 pm

First, the question asks: is |x - z| + |x| = |z|. I find this easiest to think about if I understand what the equation says about distances. Remember that |a - b| is just the distance between a and b on the number line, and |a| is the distance from a to zero. So |x - z| + |x| = |z| just says "the distance between x and z plus the distance from x to zero is equal to the distance from z to zero". How can this happen? Draw a number line, try placing x and z in different orders and on either side of zero, and you can see that this can only happen in two ways:

z < x < 0
0 < x < z

So the question is just asking- can we be sure that one of the above inequalities is true?

Before looking at the statements, we know that zy < xy < 0. What does this tell us?

-Either y is negative, and both x and z are positive, or y is positive, and both x and z are negative.

-Because zy - xy < 0, y(z-x) <0. If y is negative, then z-x must be positive, and z > x. If y is positive, z-x must be negative, and z < x (and, conversely, if z < x, y must be positive).

Okay, that was the tough part. Let's look at the statements:
1) z < x

If z < x, we know that y must be positive. If y is positive, x and z are both negative. So we know that z < x <0, and the answer to the question must be yes.

2) y > 0

If y is positive, well, this is exactly the same situation as we had with Statement 1. So again, z < x < 0, and the answer to the question must be yes.

D.

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atlantic · Mon Jun 30, 2008 3:46 am

Thanks Ian. Your explanation was of great help.

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