If y divided by z can’t equal 1, then y and z themselves are not equal. You can use logic to figure this out or you can manipulate the non-equation by multiplying both sides by z:

(1) INSUFFICIENT: Test some cases here. If x = 0, then |y| = |z|. Remember that y and z cannot be the same number! This would work, then, if y = 2 and z = –2 (or vice versa). In this case, y / z = –1. (As long as x and y are the same number but opposite in sign, you can choose any values you want, and the quotient will be –1.)

If, on the other hand, x = 1, then |1 + y| = |1 + z|. Solve for the positive version:

1 + y = 1 + z

y = z

That is an illegal response, since y can’t equal z. Try the negative version:

1 + y = –(1 + z)

1 + y = –1 – z

y + z = –2

Pick two values that make this statement true. For example, if y = –3 and z = 1, then y / z = –3. There are at least two possible values for y / z, so this statement is insufficient.

(2) INSUFFICIENT: Test some cases again. If x = 0, then |–y| = |–z|. Remember again that y and z cannot be the same number! This would work, then, if y = 2 and z = –2 (or vice versa). In this case, y / z = –1.

If, on the other hand, x = 1, then |1 – y| = |1 – z|. Since solving for the negative version worked better last time, start with the negative version this time:

1 – y = –(1 – z)

1 – y = –1 + z

2 = y + z

Pick two values that make this statement true. For example, if if y = 3 and z = –1, then y / z = –3. There are at least two possible values for y / z, so this statement is insufficient.

(1) AND (2) SUFFICIENT: For each statement alone, testing x = 0 produced the same outcome, so at the least, y and z could be “opposites” (the same number but opposite signs) and y / z = –1. Are there other cases, though, that would work for both statements?

Take a look at the full versions of the two statements that didn’t produce the illegal outcome x = y; that is, use the negative version of each:

From statement (1) x + y = –(x + z) which becomes y + z = –2x

From statement (2): x – y = –(x – z) which becomes 2x = y + z

Notice anything? There are similar terms in those equations. Remember that the problem asks about y and z, so manipulate the first equation to drop the x terms:

(1) 2x = –y – z

(2) 2x = y + z

Set the two right-hand sides equal and simplify:

–y – z = y + z

0 = 2y + 2z

0 = y + z

This final equation proves that y and z have to be opposites: if y = 2, then z = –2; if y = 3, then z = –3; and so on. In any case, then, y / z = –1.

The correct answer is (C).

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we shall fight on the beaches,

we shall fight on the landing grounds,

we shall fight in the fields and in the streets,

we shall fight in the hills;

we shall never surrender!