2+|1-x/2 |>= 5 — step-by-step solution

1) Let's solve in

ℝ

2 + | 1 - \frac{x}{2} | \geq 5

ⓘ We subtract

2

from both sides

| 1 - \frac{x}{2} | \geq 5 - 2

ⓘ Start from

5

and move back

2

on the number line:

5 - 2

3

| 1 - \frac{x}{2} | \geq 3

| A | \geq B

means that:

A \geq B

A \leq - B

2) Let's solve in

ℝ

1 - \frac{x}{2} \geq 3

ⓘ We subtract

1

from both sides

- \frac{x}{2} \geq 3 - 1

ⓘ Start from

3

and move back

1

on the number line:

3 - 1

2

- \frac{x}{2} \geq 2

ⓘ We multiply both sides by

2

- x \geq 2 \cdot 2

ⓘ A single minus sign does not cancel out, the product of a negative number by a positive number is negative:

- 2 \cdot 2

-4

-4

ⓘ We add

x

to both sides

-4 \geq x

Solution found:

x \leq -4

3) Let's solve in

ℝ

1 - \frac{x}{2} \leq -3

ⓘ We subtract

1

from both sides

- \frac{x}{2} \leq - 3 - 1

ⓘ Start from

-3

and move back

1

more on the number line, away from zero:

- 3 - 1

-4

- \frac{x}{2} \leq -4

ⓘ We multiply both sides by

2

- x \leq 2 \cdot (-4)

ⓘ Two minus signs cancel out, the product of two negative numbers is positive:

- 2 \cdot (-4)

8

8

ⓘ We add

x

to both sides

8 \leq x

Solution found:

x \geq 8

▶Solution:

x \in (-∞, -4] \cup [8, +∞)