How do you solve #|x + 1| + 1\geq 8#?

2 Answers
smendyka
Mar 18, 2018

See a solution process below:

Explanation:

First, subtract #color(red)(1)# from each side of the inequality to isolate the absolute value function while keeping the inequality balanced:

#abs(x + 1) + 1 - color(red)(1) >= 8 - color(red)(1)#

#abs(x + 1) >= 7#

The absolute value function takes any term and transforms it to its non-negative form. Therefore, we must solve the term within the absolute value function for both its negative and positive equivalent.

#-7 >= x + 1 >= 7#

Now, subtract #color(red)(1)# from each segment of the system of inequalities to solve for #x# while keeping the system balanced:

#-7 - color(red)(1) >= x + 1 - color(red)(1) >= 7 - color(red)(1)#

#-8 >= x + 0 >= 6#

#-8 >= x >= 6#

Or

#x <= -8#; #x >= 6#

Or, in interval notation:

#(-oo, -8]#; #[6, +oo)#

Gazza
Mar 18, 2018

#x>=6 or x <=-8#

Explanation:

#|x+1|+1>=8#

Let start by adding #-1#to both sides

#|x+1|+1-1>=8-1#

#|x+1|>=7#

We know either: #x+1>=7 or x + 1<=-7#

Let start with the first possibility which is:

#x + 1 >=7#

Add #-1# on both sides

#x + 1 - 1 >=7-1#

#x >= 6#

Now the second possibility

#x+1<=-7#

Add #-1# on both sides

#x + 1 - 1 <= -7 -1#

#x <=-8#

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