I got no solution??

Question

Let f(x)= |2x-3| and g(x)= |-7+2x|. Find all x for which f(x)=g(x)

 

After solving I got no solution but it does not seem right. 

Answer 1

f(x) = |2x − 3| = 2|x − 3/2|, g(x) = |2x − 7| = 2|x − 7/2|.  They are just the vertically scaled and horizontally shifted versions of the function |x|.

 

Sketching f(x) and g(x) would help you understand the solution.

 

f(x) = |2x − 3| =  2x − 3 for x ≥ 3/2

                         −2x + 3 for x < 3/2

 

f(x) = |2x − 7| = 2x − 7 for x ≥ 7/2
                        −2x + 7 for x < 7/2

 

By inspection, the solution can only exist in the interval 3/2 ≤ x < 7/2 where

2x − 3 = −2x + 7

 

So the solution is 

x = 2.5

Answer 2

Four possibilities:

Both arguments are positive

2x - 3 = -7 + 2x

-3 = -7 Boo!

First positive second negative

2x - 3 = -(-7 + 2x)

2x - 3 = 7 - 2x

4x = 10

x = 2.5  Yaaa!

Both are negative

-(2x - 3) = -(-7 + 2x)

-2x + 3 = 7 - 2x

3 = 7  Boo!

First is negative and second is positive

-(2x - 3) = -7 + 2x

-2x + 3 = -7 + 2x

-4x = -10

x = 2.5  Again!