graph the solution set of the compound inequality: |x+1|<4 or |x+1|>=6

Question

 Please graph on a number line.

Answer 1

Solve the individual inequalities as follows:

 

|x+1| < 4

-4 < x + 1 < 4

-5 < x < 3, or x is in (-5, 3)

 

|x+1| >= 6

x + 1 < -6 OR x + 1 > 6

x < -7 OR x > 5, or x is in (-inf, -7] U [5, inf)

 

Since the two original inequalities are connected by an OR, we want the union of the solution sets. This is

{(-inf, -7] U [5, inf)} U (-5, 3),

which simplifies to

 

(-inf, -7] U (-5, 3) U [5, inf)

 

So on the number line,

1) Put a bracket or filled circle on -7 and shade everything to its left;2) Put parentheses or open circles on -5 and 3 and shade everything between; and,

3) Put a bracket or filled circle on 5 and shade everything to its right.

 

Answer 2

|x+1|<4

-4 < x+1<4

-4-1 < x < 4-1

-5 <x<3

All numbers between -5 and 3 not including the endpoints:

In interval notation x is  (-5, 3)

 

 

|x+1 ≥ 6

6 ≤ x+1 ≤ -6

7 ≤ x ≤ -7

All numbers larger than or equal to 7 or less than or equal to -7: 

In interval notation x is (-∞, -7] U [7, ∞)

 

 

<-------←*---o-------|----o-------*→----------->

             -7   -5        0     3          7

 

Solid dot at -7 with arrow running to the left

open circle at -5 and 3 and line darkened between those points

Solid dot at 7 with arrow running to the right