Let x and y be integers such that |x| \leq 4 and |y| \leq 4. How many distinct pairs of in… — ACT

Let $x$ and $y$ be integers such that $|x| \leq 4$ and $|y| \leq 4$ . How many distinct pairs of integers $(x, y)$ satisfy the inequality $||x| - |y|| \geq 2$ ?

Cevap: 36

Cevap

There are 36 distinct pairs of integers

(x, y)

that satisfy the inequality.

By analyzing the possible values for

|x|

and

|y|

within the set

\{0, 1, 2, 3, 4\}

, we identify the 12 pairs that satisfy the inequality

||x| - |y|| \geq 2

. When mapping these absolute values back to the actual integer coordinates, we account for the single option when a coordinate is 0 and the two positive/negative options when a coordinate is non-zero. Summing these possibilities yields exactly 36 distinct pairs.

Adım Adım Çözüm

Determine the range of absolute values for the integers.

The possible values for

|x|

and

|y|

are

\{0, 1, 2, 3, 4\}

Since

x

and

y

are integers satisfying

-4 \leq x, y \leq 4

, their absolute values must be non-negative integers up to 4.

Find the pairs of absolute values

(|x|, |y|)

that satisfy the inequality.

The valid pairs are:

(0, 2), (0, 3), (0, 4), (1, 3), (1, 4), (2, 0), (2, 4), (3, 0), (3, 1), (4, 0), (4, 1), (4, 2)

We test all combinations of

u, v \in \{0, 1, 2, 3, 4\}

such that

|u - v| \geq 2

Calculate the number of integer coordinate pairs

(x, y)

corresponding to each absolute value pair.

Pairs with one zero value yield 2 integer solutions (e.g.,

u=0, v=2 \Rightarrow (0, 2), (0, -2)

). Pairs with both non-zero values yield 4 integer solutions (e.g.,

u=1, v=3 \Rightarrow (1, 3), (1, -3), (-1, 3), (-1, -3)

An absolute value of 0 corresponds to only 1 integer (

0

), whereas any positive absolute value

k

corresponds to 2 integers (

k

and

-k

Sum the number of integer pairs for all valid cases.

Total pairs =

(3 \times 2) + (3 \times 2) + (6 \times 4) = 6 + 6 + 24 = 36

There are 3 pairs with

u=0

(

6

solutions), 3 pairs with

v=0

(

6

solutions), and 6 pairs with both

u, v > 0

(

24

solutions).

Anahtar Kavram

Solving nested absolute value inequalities with integer constraints and counting solution pairs systematically.

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