Solve the following systems of linear in equations: 1 ≤ |x – 2| �

1.	Solve the following systems of linear in equations: 1 ≤ \|x – 2\| ≤ 3
Answer» 1 ≤ \|x - 2\| and \|x - 2\| ≤ 3 When, \|x - 2\| ≥ 1 Then, x – 2 ≤ -1 and x -2 ≥ 1 Now when, x – 2 ≤ - 1 Adding 2 to both the sides in above equation x – 2 + 2 ≤ -1 + 2 x ≤ 1 Now when, x – 2 ≥ 1 Adding 2 to both the sides in above equation x – 2 + 2 ≥ 1 + 2 x ≥ 3 For \|x – 2\| ≥ 1: x ≤ 1 or x ≥ 3 When, \|x - 2\| ≤ 3 Then, x – 2 ≥ - 3 and x – 2 ≤ 3 Now when, x – 2 ≥ -3 Adding 2 to both the sides in above equation x – 2 + 2 ≥ -3 + 2 x ≥ -1 Now when, x – 2 ≤ 3 Adding 2 to both the sides in above equation x – 2 + 2 ≤ 3 + 2 x ≤ 5 For \|x – 2\| ≤ 3: x ≥ -1 or x ≤ 5 Combining the intervals: x ≤ 1 or x ≥ 3 and x ≥ -1 or x ≤ 5 Merging the overlapping intervals: -1 ≤ x ≤ 1 and 3 ≤ x ≤ 5 Therefore, x ϵ [-1 ,1] Ս [3, 5]

Solve the following systems of linear in equations: 1 ≤ |x – 2| ≤ 3

Answer»

1 ≤ |x - 2| and |x - 2| ≤ 3

When,

|x - 2| ≥ 1

Then,

x – 2 ≤ -1 and x -2 ≥ 1

Now when,

x – 2 ≤ - 1

Adding 2 to both the sides in above equation

x – 2 + 2 ≤ -1 + 2

x ≤ 1

Now when,

x – 2 ≥ 1

Adding 2 to both the sides in above equation

x – 2 + 2 ≥ 1 + 2

x ≥ 3

For |x – 2| ≥ 1: x ≤ 1 or x ≥ 3

When,

|x - 2| ≤ 3

Then,

x – 2 ≥ - 3 and x – 2 ≤ 3

Now when,

x – 2 ≥ -3

Adding 2 to both the sides in above equation

x – 2 + 2 ≥ -3 + 2 x ≥ -1

Now when,

x – 2 ≤ 3

Adding 2 to both the sides in above equation

x – 2 + 2 ≤ 3 + 2

x ≤ 5

For |x – 2| ≤ 3: x ≥ -1 or x ≤ 5

Combining the intervals:

x ≤ 1 or x ≥ 3 and x ≥ -1 or x ≤ 5

Merging the overlapping intervals:

-1 ≤ x ≤ 1 and 3 ≤ x ≤ 5

Therefore,

x ϵ [-1 ,1] Ս [3, 5]