Name: Randall

Who is asking: Teacher
Level: Secondary

Question:
I don't seem to be able to solve this:

| n + 4 | + | 3 - 2n | = 16 Can you help?



Hi Randall,

I think that the most straightforward way to approach this problem is with a case argument. The cases come from the fact that

if a is non-negative then | a | = a and if a is negative then | a | = - a

There are four cases, depending on the sign of n + 4 and 3 - 2n.

Case 1: n + 4 >= 0 and 3 - 2n >= 0

| n + 4 | + | 3 - 2n | = 16
n + 4 + 3 - 2n = 16
n = - 9

Check:
| -9 + 4 | + | 3 -2(-9) |
= 5 + 21 = 26 which is not 16.

Hence n = -9 is not a solution. Case 2: n + 4 >= 0 and 3 - 2n < 0 | n + 4 | + | 3 - 2n | = 16
n + 4 - 3 + 2n = 16
3n = 15
n = 5

Check:
| 5 + 4 | + | 3 -2(5) |
= 9 + 7 = 16.

Hence n = 5 is a solution. Case 3: n + 4 < 0 and 3 - 2n >= 0 | n + 4 | + | 3 - 2n | = 16
- n - 4 + 3 - 2n = 16
-3 n = 17
n =  -17/3

Check:
|  -17/3 + 4 | + | 3 -2(  -17/3) |
= 5/3 + 43/3 = 16.

Hence n =  -17/3 is a solution. Case 4: n + 4 < 0 and 3 - 2n < 0 | n + 4 | + | 3 - 2n | = 16
- n - 4 - 3 + 2n = 16
n = 23

Check:
| 23 + 4 | + | 3 -2(23) |
= 27 + 43 = 70 which is not 16.

Hence n = 23 is a solution.

Thus there are two solutions, n = 5 and n =  -17/3

Cheers,
Penny

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