Three distinct integers, a, b, and c, lie on a standard number line such that a < b < c. T… — ACT

Three distinct integers, $a$ , $b$ , and $c$ , lie on a standard number line such that $a < b < c$ . The distance between $a$ and $b$ is $3$ times the distance between $b$ and $c$ . If $|a| = 15$ , $|c| = 7$ , and $b < 0$ , what is the value of $b$ ?

Answer: -9

Answer

The value of

b

-9

The correct answer is

-9

. By interpreting

|a| = 15

and

|c| = 7

with the constraint

a < b < c

, we find

a = -15

. Testing the possible values for

c

, when

c = -7

, we set up the distance equation

b - (-15) = 3(-7 - b)

, which simplifies to

b + 15 = -21 - 3b

, giving

4b = -36

and

b = -9

. This satisfies all constraints, including

b

being a negative integer.

Step-by-Step Solution

Find the possible coordinates for

a

and

c

based on their absolute values.

a \in \{-15, 15\}

and

c \in \{-7, 7\}

The absolute value of a number represents its distance from zero, so

|x| = d \implies x = \pm d

Use the ordering condition

a < b < c

to eliminate invalid combinations.

a = -15

and

c \in \{-7, 7\}

Since

a

must be less than

c

a

cannot be

15

because both possible values of

c

(

-7

and

7

) are less than

15

Set up an equation representing the distance relationship on the number line.

b + 15 = 3(c - b) \implies 4b = 3c - 15

For points on a number line ordered

a < b < c

, the distance between

a

and

b

b - a

, and the distance between

b

and

c

c - b

Substitute each possible value of

c

and solve for

b

to find the one that yields a negative integer.

For

c = -7

b = -9

c = 7

b = 1.5

, which is not an integer. If

c = -7

b = -9

, which is a negative integer, satisfying all given conditions.

Key Concept

Using absolute values and relative order to determine integer positions and distances on a number line.

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Key Concept

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