Find all those numbers x on the number line that are twice as far from 2 as they are from 10.
|x-2|=2|x-10|
If x>10, x-10 and x-2 are both positive, so we have
x-2 = 2(x-10)
x=18
Now check for another answer when 2<x<10 or x<2
Thank you Steve
To find the numbers that are twice as far from 2 as they are from 10, we can use the concept of the distance between two numbers on a number line.
Let's first represent the numbers on the number line. We have 2 and 10.
To find the numbers that are twice as far from 2 as they are from 10, we need to find the numbers that satisfy the following condition:
Distance from the number to 2 = 2 * (Distance from the number to 10)
Let's denote the number as x.
The distance from x to 2 is |x - 2|, and the distance from x to 10 is |x - 10|.
Using the above condition, we can set up the equation:
|x - 2| = 2 * |x - 10|
Now, let's solve the equation step-by-step:
1. When x is greater than or equal to 10:
In this case, the equation becomes:
x - 2 = 2 * (x - 10)
Solve for x:
x - 2 = 2x - 20
x - 2x = -20 + 2
-x = -18
x = 18
Therefore, for values of x that are greater than or equal to 10, the only solution is x = 18.
2. When x is between 2 and 10:
In this case, the equation remains the same:
|x - 2| = 2 * |x - 10|
To solve this equation, we need to consider two cases:
a. x - 2 = 2 * (x - 10)
Solve for x:
x - 2 = 2x - 20
x - 2x = -20 + 2
-x = -18
x = 18
However, this value of x does not satisfy our assumption that x is between 2 and 10. Therefore, it is not a valid solution in this case.
b. x - 2 = -2 * (x - 10)
Solve for x:
x - 2 = -2x + 20
x + 2x = 20 + 2
3x = 22
x = 22/3
Therefore, for values of x that are between 2 and 10, the solution is x = 22/3 or approximately 7.33.
To summarize, the numbers on the number line that are twice as far from 2 as they are from 10 are:
- For x >= 10: x = 18
- For 2 < x < 10: x = 22/3 or approximately 7.33