1<|x-1|<= 3 — step-by-step solution

1) We have the following 2 conditions:

[1]:

1 < | x - 1 |

[2]:

| x - 1 | \leq 3

2) Let's solve for x:

1 < | x - 1 |

[1]

| x - 1 | > 1

| A | > B

means that:

A > B

A < - B

x - 1 > 1

ⓘ We add

1

to both sides

x > 1 + 1

ⓘ Start from

1

and move forward

1

on the number line:

1 + 1

2

Solution obtained [1]:

x > 2

x - 1 < -1

ⓘ We add

1

to both sides

x < - 1 + 1

ⓘ Start from

-1

and move forward

1

on the number line, up to zero:

- 1 + 1

0

Solution found [1]:

x < 0

3) Solving for x:

| x - 1 | \leq 3

[2] using

x \in (-∞, 0) \cup (2, +∞)

| x - 1 | \leq 3

| A | \leq B

is equivalent to

- B \leq A \leq B

-3 \leq x - 1 \leq 3

We add

1

to all three parts of the inequality

-2 \leq x \leq 4

Solution found [2]:

x \in [-2, 4]

Let's create the table of the different solutions to find the overall solution:

Intersecting

x \in (-∞, 0) \cup (2, +∞)

and

x \in [-2, 4]

gives

x \in [-2, 0) \cup (2, 4]

▶Solution:

x \in [-2, 0) \cup (2, 4]