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

2 Answers
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)

Mar 18, 2018

x>=6 or x <=-8

Explanation:

|x+1|+1>=8

Let start by adding -1to 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