On a number line, point M has coordinate -14 and point N has coordinate 10. Point P is loc… — ACT

On a number line, point $M$ has coordinate $-14$ and point $N$ has coordinate $10$ . Point $P$ is located on the number line such that the distance between $M$ and $P$ is three times the distance between $N$ and $P$ . If the coordinate of $P$ is positive, what is the sum of all possible coordinates of $P$ ?

A
$4$
B
$8$
C
$22$
$26$ Answer
E
$34$

Answer

The sum of all possible coordinates of

P

26

The distance between any two points

x

and

y

on a number line is given by

|x - y|

. Thus, the distance from point

P(p)

M(-14)

|p - (-14)| = |p + 14|

, and the distance from

P(p)

N(10)

|p - 10|

. Since the distance to

M

is three times the distance to

N

, we write

|p + 14| = 3|p - 10|

. To solve this absolute value equation, we check two cases. In the first case, we have

p + 14 = 3(p - 10)

, which simplifies to

2p = 44

, or

p = 22

. In the second case, we have

p + 14 = -3(p - 10)

, which simplifies to

4p = 16

, or

p = 4

. Both coordinates are positive, satisfying the condition given in the problem. The sum of these possible coordinates is

22 + 4 = 26

Step-by-Step Solution

Set up the algebraic representation of the distances between the points on the number line using absolute value.

The distance between point

M(-14)

and point

P(p)

|p - (-14)| = |p + 14|

. The distance between point

N(10)

and point

P(p)

|p - 10|

The distance between two points

a

and

b

on a standard number line is always expressed as the absolute value of their difference,

|a - b|

Formulate the equation representing the relationship between the two distances.

|p + 14| = 3|p - 10|

The problem states that the distance from

M

P

is three times the distance from

N

P

Solve the absolute value equation by analyzing both positive and negative cases.

Case 1:

p + 14 = 3(p - 10) \implies p + 14 = 3p - 30 \implies 2p = 44 \implies p = 22

.
Case 2:

p + 14 = -3(p - 10) \implies p + 14 = -3p + 30 \implies 4p = 16 \implies p = 4

An equation of the form

|A| = |B|

is solved by evaluating the two distinct possibilities:

A = B

and

A = -B

Check if both solutions satisfy the condition of being positive, and sum them.

Both

22

and

4

are positive, so they are both valid coordinates. Their sum is

22 + 4 = 26

The question specifies that the coordinate of

P

must be positive, and asks for the sum of all such coordinates.

Key Concept

Representing distances on a number line using absolute value and solving absolute value equations.

Estimated Time:1m 30s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic