Let a and b be integers. On a standard number line, a = -14, and the distance between a an… — ACT

Let $a$ and $b$ be integers. On a standard number line, $a = -14$ , and the distance between $a$ and $b$ is $23$ . If $b > 0$ , what is the value of $|2b - |a||$ ?

Answer: 4

Answer

The value of

|2b - |a||

is 4.

The distance between

a = -14

and

b

on the number line is

|b - (-14)| = |b + 14| = 23

. Solving for a positive value of

b

gives

b + 14 = 23

, so

b = 9

. The absolute value of

a

|-14| = 14

. Substituting these values into

|2b - |a||

yields

|2(9) - 14| = |18 - 14| = 4

Step-by-Step Solution

Determine the value of integer

b

using the distance relationship on the number line.

The distance formula

|b - (-14)| = 23

simplifies to

|b + 14| = 23

. Since

b > 0

b = 9

The distance between two points

x

and

y

on a number line is given by

|x - y|

Calculate the absolute value of

a

|a| = |-14| = 14

The absolute value of a negative number is its positive magnitude.

Substitute

b = 9

and

|a| = 14

into the expression

|2b - |a||

|2(9) - 14| = |18 - 14| = 4

Perform operations inside the outer absolute value first, then take the absolute value of the result.

Key Concept

Absolute Value as Distance on a Number Line

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