On a standard number line, the coordinates of three distinct points A, B, and C are repres… — ACT

On a standard number line, the coordinates of three distinct points $A$ , $B$ , and $C$ are represented by the integers $a$ , $b$ , and $c$ , respectively, such that $a < b < c$ . The distance between each point and the origin is represented by its absolute value, and these distances satisfy the inequality $|a| < |b| < |c|$ . If the product of the three coordinates is negative ( $abc < 0$ ) and the sum of their absolute values is $20$ , what is the maximum possible value of the coordinate $b$ ?

A
6
B
8
9Answer
D
10
E
18

Answer

The correct answer is 9. By analyzing the signs of the coordinates, we determine that only

a

is negative, so

a < 0 < b < c

. Since

a

is a negative integer, the minimum value of

|a|

is 1. Under the constraint

|a| < |b| < |c|

and

|a| + |b| + |c| = 20

, we let

x = -a

y = b

, and

z = c

. We have

x + y + z = 20

with

1 \le x < y < z

. Since

z > y

, we get

20 = x + y + z > 1 + 2y

, which implies

2y < 19

, or

y < 9.5

. The maximum integer value for

y

(which is

b

) is 9, achieved when the coordinates are -1, 9, and 10.

Step-by-Step Solution

Determine the signs of the coordinates based on the given constraints.

The sign configuration must be

a < 0 < b < c

Since the product

abc < 0

, either all three coordinates are negative or exactly one is negative. If all three were negative (

a < b < c < 0

), their absolute values would satisfy

|a| > |b| > |c|

, which contradicts

|a| < |b| < |c|

. Thus, exactly one coordinate (

a

) is negative, and the other two (

b

and

c

) are positive.

Translate the absolute values and sum constraint into algebraic terms.

-a + b + c = 20

, where

a < 0

and

b, c > 0

For a negative number

a

, the absolute value

|a| = -a

. For positive numbers

b

and

c

|b| = b

and

|c| = c

. Therefore, the sum

|a| + |b| + |c| = 20

simplifies to

-a + b + c = 20

Set up inequalities to bound the coordinate of the middle point.

1 \le -a < b < c

Since

a

is a non-zero negative integer, the smallest possible value for its absolute value

|a| = -a

1

. The condition

|a| < |b| < |c|

then becomes

1 \le -a < b < c

Use the constraints to find the maximum value of

b

The maximum possible value of

b

9

Let

x = -a

y = b

, and

z = c

. We have

x + y + z = 20

with

1 \le x < y < z

. Since

z > y

, we have

20 = x + y + z > x + 2y \ge 1 + 2y

. This simplifies to

19 > 2y

, or

y < 9.5

. Since

y

must be an integer, the maximum possible value for

y

(which is

b

) is

9

. We can verify this maximum by setting

a = -1

b = 9

, and

c = 10

, which perfectly satisfies all conditions.

Key Concept

Analyzing coordinate signs and absolute value inequalities on a number line to perform optimization under integer constraints.

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