SOLUTION: Rewrite without absolute value sign for different values of x: y= −|2x+5|+|2x−5| if x<−2.5; if x>2.5; if −2.5≤x≤2.5

Algebra ->  Absolute-value -> SOLUTION: Rewrite without absolute value sign for different values of x: y= −|2x+5|+|2x−5| if x<−2.5; if x>2.5; if −2.5≤x≤2.5       Log On


   

Question 1140329: Rewrite without absolute value sign for different values of x:
y= −|2x+5|+|2x−5|
if x<−2.5;
if x>2.5;
if −2.5≤x≤2.5

Answer by greenestamps(13203) About Me  (Show Source):
You can put this solution on YOUR website!


abs%28A%29+=+A if A >= 0;
abs%28A%29+=+-A if A < 0.

(1) If x < -2.5, then...

2x+5 and 2x-5 are both <0, so

abs%282x%2B5%29+=+-%282x%2B5%29
-abs%282x%2B5%29+=+2x%2B5
abs%282x-5%29+=+-%282x-5%29+=+-2x%2B5
-abs%282x%2B5%29%2Babs%282x-5%29+=+%282x%2B5%29%2B%28-2x%2B5%29+=+10

(2) If x >= 2.5, then...

2x+5 and 2x-5 are both >0, so

abs%282x%2B5%29+=+2x%2B5
-abs%282x%2B5%29+=+-%282x%2B5%29+=+-2x-5
abs%282x-5%29+=+2x-5
-abs%282x%2B5%29%2Babs%282x-5%29+=+%28-2x-5%29%2B%282x-5%29+=+-10

(3) If -2.5 < x <= 2.5, then...

2x+5 >0 and 2x-5 < 0, so

abs%282x%2B5%29+=+2x%2B5
-abs%282x%2B5%29+=+-%282x%2B5%29+=+-2x-5
abs%282x-5%29+=+-%282x-5%29+=+-2x%2B5%29
-abs%282x%2B5%29%2Babs%282x-5%29+=+%28-2x-5%29%2B%28-2x%2B5%29+=+-4x

Summary:

y = 10; if x < -2.5
y = -4x; if -2.5 <= x <= 2.5
y = -10; if x > 2.5

A graph....

graph%28400%2C400%2C-5%2C5%2C-15%2C15%2C-abs%282x%2B5%29%2Babs%282x-5%29%29