I need help in calculus! rewrite absolute value functions into piecewise functions!!?

Question

i understand how to do the shorter ones like y = | x - 2|

but what about

y = |9 - x| +7

and

y = 5 |3x - 4|

I know how to find the boundaries.

Answer 1

The definition of the absolute value does not change!

The definition is simply

|x| = x when x>=0

|x| = -x when x<0

y = 9-x + 7 for 9-x >= 0

y = -(9-x) + 7 for 9-x < 0

Now simplify (oops made a mistake earlier, if you didn't catch it, however it has been corrected)

y = 16-x  for x <= 9

y = x-2  for x > 9

Same applies for

y = 5 |3x - 4|

y = 5(3x-4) for 3x-4 >= 0

y = 5(-(3x-4)) for 3x-4 < 0

Simplify

y = 15x-20 for x >= 4/3

y = -15x+20 for x < 4/3