How do you solve this compound inequality #2x + 6\geq 12 or 2x + 6\leq - 4#?

We have a Compound Inequality :-

#color(green)(2x + 6 >= 12 or 2x + 6 <= -4)#

We will consider our compound inequality as two individual parts to start with:

#color(green)(2x + 6 >= 12# . . . Inequality.1

#color(green)(2x + 6 <= -4)# . . . Inequality.2

We will start with . . . Inequality.2 first:

#color(green)(2x + 6 <= -4)# . . .Inequality.2

Add #-6# to both sides of our inequality:

#color(green)(2x + cancel (+6) cancel(- 6) <= -4 - 6#

#color(green)(rArr 2x <= -10#

Divide both sides of the inequality by #2#

#color(green)(rArr (cancel 2x)/cancel 2 >= -10/2#

#color(green)(rArr x >= -5# #color(red)( ... Intermediate.Answer.1)#

Next we will consider . . . Inequality.1 :

#color(green)(2x + 6 >= 12# . . . Inequality.1

Add #-6# to both sides of our inequality:

#color(green)(2x + cancel (+6) cancel(- 6) >= 12 - 6#

#color(green)(rArr 2x >= 6#

Divide both sides of the inequality by #2#

#color(green)(rArr (cancel 2x)/cancel 2 >= 6/2#

#color(green)(rArr x >= 3# #color(red)( ... Intermediate.Answer.2)#

Using our intermediate answers 1 and 2 we can write our solutions as:

#color(red)( x <= -5 or x >= 3)#

Using Interval Notation:

#color(blue)((-oo, -5] uu [3, oo)#

Refer to the graph below for a better understanding:

I hope you find this solution useful.