On the real number line, the distance between x and 3 is equal to twice the distance betwe… — GRE General Test

On the real number line, the distance between $x$ and $3$ is equal to twice the distance between $x$ and $-9$ . What is the sum of all possible real values of $x$ ?

$-26$ Answer
B
$-21$
C
$-14$
D
$-5$
E
$22$

Answer

The sum of all possible real values of

x

-26

The distance between

x

and

a

on the real number line is expressed as

|x - a|

. Thus, the condition translates to

|x - 3| = 2|x - (-9)|

, which simplifies to

|x - 3| = 2|x + 9|

. Setting up the two algebraic cases gives

x - 3 = 2(x + 9)

, leading to

x = -21

, and

x - 3 = -2(x + 9)

, leading to

x = -5

. Adding these two solutions yields

(-21) + (-5) = -26

Step-by-Step Solution

Translate the geometric statement on the number line into an absolute value equation.

The distance between

x

and

3

|x - 3|

, and the distance between

x

and

-9

|x - (-9)| = |x + 9|

. Therefore,

|x - 3| = 2|x + 9|

Distance between two points

a

and

b

on the number line is given by

|a - b|

Solve Case 1 where the expressions inside the absolute values have the same sign.

x - 3 = 2(x + 9) \implies x - 3 = 2x + 18 \implies x = -21

Removing absolute values with identical signs gives a linear equation in

x

Solve Case 2 where the expressions inside the absolute values have opposite signs.

x - 3 = -2(x + 9) \implies x - 3 = -2x - 18 \implies 3x = -15 \implies x = -5

Removing absolute values with opposite signs accounts for the second possible geometric location.

Sum the valid real solutions.

(-21) + (-5) = -26

The question asks for the sum of all possible real values of

x

Key Concept

Absolute Value as Distance on the Real Number Line

Question

Answer

Step-by-Step Solution

Key Concept

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