Use interval notation to represent all values of x satisfying the given conditions y=9-|x/2+2| and y is at most 4

Answer 1

y = 9 - | x/2 + 2|

y-9 = -|x/2 + 2|

9 - y = | x/2 +2}

So then

x/2 +2 = 9 -y OR x/2 +2 = y - 9

In the former case:

x/2 - 7 = -y

y = (-1/2)x + 7

in the latter case:

y = (1/2)x + 11

since y<=4

(-1/2)x + 7 <= 4 or (1/2)x + 11<=4

(-1/2)x <= -3 or (1/2)x <= -7

x >= 6 or x < = -14

(-infinity, -14) U (6,infinity)

CHECK:

Note that for x=-14 and x=6, y = 9 - |x/2+2| =4

and for x<-14 or x>6, y<4

But for x=0 (not in the solution), y = 9 - |x/2+2| = 7 > 4

so the solution checks

(-infinity, -14) U (6,infinity)