How do you solve -3+2abs(n-9)=1?

2 Answers
Stefan V.
Aug 1, 2015

n=7 or n=11

Explanation:

First, isolate the modulus on one side of the equation by adding 3 to both sides of the equation and dividing everything by 2.

- color(red)(cancel(color(black)(3))) + color(red)(cancel(color(black)(3))) + 2|n-9| = 1 + 3

(color(red)(cancel(color(black)(2))) |n-9|)/color(red)(cancel(color(black)(2))) = 4/2

|n-9| = 2

Now, this equation can produce 2 distinct values for n depending on which condition is true

  • If (n-9)<0, then you have

|n-9| = -(n-9) = -n + 9

This means that the equation becomes

-n + 9 = 2 => n = color(green)(7)

  • If (n-9)>0, then you have

|n-9| = n-9

This means that n will be equal to

n-9 = 2 => n = color(green)(11)

Jim H
Aug 1, 2015

There is a geometric way to think about this.

Explanation:

Starting with -3+2abs(n-9) = 1,

we can see that we must have:

2abs(n-9) = 4, so

abs(n-9) = 2

On the number line, this means that the distance between n and 9 is 2.
There are two numbers that are 2 away from 9: one on the left of 9 at 9-2 and the other on the right at 9+2.

So, we get:

n = 9-2" " or " "n=9+2

and

n=7" " or " "n=11

Related questions
See all questions in Absolute Value Equations
Impact of this question
1746 views around the world
You can reuse this answer
Creative Commons License