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

Question

Eli Assouline · Accepted Answer

#n=7# or #n=11#

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

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

This means that the equation becomes

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

#|n-9| = n-9#

This means that #n# will be equal to

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

Savannah Anderson · Answer

There is a geometric way to think about this.

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#

Avery Alexander · Answer

undefined To solve the equation $ -3 + 2|n - 9| = 1 $, you can follow these steps:

1. Add 3 to both sides to isolate the absolute value term.
2. $ -3 + 3 + 2|n - 9| = 1 + 3 $
3. $ 2|n - 9| = 4 $
4. Divide both sides by 2 to isolate the absolute value term.
5. $ \frac{2|n - 9|}{2} = \frac{4}{2} $
6. $ |n - 9| = 2 $

Now, the absolute value can be either positive or negative, so you'll have two cases:

Case 1: $ n - 9 = 2 $
1. Add 9 to both sides.
2. $ n = 11 $

Case 2: $ -(n - 9) = 2 $
1. Multiply both sides by -1 to remove the negative sign.
2. $ n - 9 = -2 $
3. Add 9 to both sides.
4. $ n = 7 $

So, the solutions to the equation are $ n = 11 $ and $ n = 7 $.