Show that if |x+3|<1/2, then |4x+13|<3?

Question

so far i got

|x+3|<1/2

x+3<1/2 -x-3<1/2

x<-5/2 -x<7/2

x>-7/2

-7/2<x<-5/2

-14<4x<-10

-1<4x+13<3

Answer 1

|x + 3| &lt; 1/2

Multiply both sides by 4:

4|x + 3| &lt; 4(1/2)

|4(x + 3)| &lt; 2

|4x + 12| &lt; 2

|4x + 12| + |1| &lt; 2 + |1| = 3

But we have by the triangle inequality that:

|(4x + 12) + 1| = |4x + 13| &lt; |4x + 12| + |1| &lt; 3

Therefore, by transitivity we have that:

|4x + 13| &lt; 3

Hope this helps!