Four distinct integers, p, q, r, and s, are represented on a standard number line. The dis… — ACT

Four distinct integers, $p$ , $q$ , $r$ , and $s$ , are represented on a standard number line. The distance between $p$ and $q$ is 3, the distance between $q$ and $r$ is 4, and the distance between $r$ and $s$ is 5. What is the minimum possible distance between $p$ and $s$ ?

Answer: 2

Answer

The minimum possible distance between

p

and

s

is 2.

By setting

p = 0

as a reference point, the coordinate for

q

must be either

3

-3

. Assuming

q = 3

by symmetry,

r

must be either

-1

7

because it is at a distance of 4 from

q

. From

r = 7

, a distance of 5 leads to

s

being at

2

12

. From

r = -1

, a distance of 5 leads to

s

being at

4

-6

. All of these positions result in four distinct integers. The distances between

p

and

s

are the absolute values of their coordinates, which are

12

2

6

, and

4

. The minimum of these distances is

2

Step-by-Step Solution

Establish a coordinate system on the number line.

Let

p

be at position 0. Since the distance between

p

and

q

is 3,

q

is at either

3

-3

. By symmetry, we assume

q = 3

Setting one point at the origin simplifies the relative distance calculations for all other points.

Find the possible coordinates of

r

Since the distance between

q

and

r

is 4,

r

is at

3 - 4 = -1

3 + 4 = 7

The absolute value equation

|q - r| = 4

has two solutions for

r

given

q = 3

Find the possible coordinates of

s

for each case of

r

r = 7

, then

s

can be at

7 - 5 = 2

7 + 5 = 12

. If

r = -1

, then

s

can be at

-1 - 5 = -6

-1 + 5 = 4

. All generated sets contain distinct values, satisfying the requirement.

The absolute value equation

|r - s| = 5

has two solutions for

s

for each candidate coordinate of

r

Determine the minimum distance between

p

and

s

The possible values for the distance

|p - s|

are

|0 - 2| = 2

|0 - 12| = 12

|0 - (-6)| = 6

, and

|0 - 4| = 4

. The minimum value is 2.

Comparing all possible valid configurations ensures we find the true minimum distance.

Key Concept

Representing distances between points on a number line using absolute values and resolving configurations for distinct integers.

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