On a number line, point A is located at 10 and point B is located at 2. If point P, with c… — GRE General Test

On a number line, point $A$ is located at $10$ and point $B$ is located at $2$ . If point $P$ , with coordinate $x > 0$ , is three times as far from point $A$ as it is from point $B$ , what is the value of $x$ ?

Answer: 4

Answer

The distance from

P(x)

A(10)

|x - 10|

and to

B(2)

|x - 2|

. Setting

|x - 10| = 3|x - 2|

gives two equations:

x - 10 = 3x - 6

, which yields

x = -2

, and

x - 10 = -3x + 6

, which yields

x = 4

. Since

x > 0

, the correct value is 4.

Step-by-Step Solution

Formulate the distance relationship using absolute value notation.

|x - 10| = 3|x - 2|

The distance between two points

u

and

v

on the real number line is given by

|u - v|

Split the absolute value equation into two linear equations representing possible cases.

x - 10 = 3(x - 2)

x - 10 = -3(x - 2)

The equality

|a| = |b|

implies

a = b

a = -b

Solve each case algebraically.

Case 1 gives

x - 10 = 3x - 6 \implies 2x = -4 \implies x = -2

. Case 2 gives

x - 10 = -3x + 6 \implies 4x = 16 \implies x = 4

Standard linear equation solving.

Apply the given domain condition

x > 0

x = 4

The solution

x = -2

is negative and therefore violates the condition

x > 0

Key Concept

Distance on a number line represented by absolute value equations

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