Official Guide Explanation:
Problem Solving #152




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Solution and Metadata

Question: 152
Page: 82
Difficulty: 6 (Moderately Difficult)
Category 1: Algebra > Linear Equations-Two Unk >
Category 2: Algebra > Absolute Value >

Explanation: First, consider the number and range of possibilities for the second equation. The absolute value of y must be less than or equal to 12, so if we're looking only for pairs of values including two integers, we should start by considering every integer for y between -12 and 12, inclusive.

Try a few of the possible values to see what might work. If y = 12, then:

2x + 12 = 12

2x = 0

x = 0

That's acceptable.

If y = 11:

2x + 11 = 12

2x = 1

x = (1/2)

Not acceptable. The same applies for negatives. If y= - 12, 2x = 24, and x is an integer. But if y= - 11, 2x = 23, and x is not an integer.

The rule, then, is that all even values for y result in an integer value for x. Thus, we need to know how many even values are acceptable for y. Between -12 and 12, we have:

-12, -10, -8, -6, -4, -2, 0, 2, 4, 6, 8, 10, and 12

Don't forget zero, and you have 13 values, choice (D).

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