9-|2x+6|<= 5 — step-by-step solution

1) Let's solve in

ℝ

9 - | 2 x + 6 | \leq 5

ⓘ We subtract

9

from both sides

- | 2 x + 6 | \leq 5 - 9

ⓘ Start from

5

and move back

9

on the number line, going below zero:

5 - 9

-4

- | 2 x + 6 | \leq -4

We divide both sides by

-1

; since this number is negative, we reverse the inequality sign

| A | \geq B

is equivalent to:

A \geq B

A \leq - B

2) Let's solve in

ℝ

2 x + 6 \geq 4

ⓘ We subtract

6

from both sides

2 x \geq 4 - 6

ⓘ Start from

4

and move back

6

on the number line, going below zero:

4 - 6

-2

2 x \geq -2

ⓘ Both sides are divided by

2

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

ⓘ We simplify the fraction by dividing the numerator and the denominator by

2

, their greatest common divisor:

- \frac{2}{2}

-1

Solution found:

x \geq -1

3) Let's solve in

ℝ

2 x + 6 \leq -4

ⓘ We subtract

6

from both sides

2 x \leq - 4 - 6

ⓘ Start from

-4

and move back

6

more on the number line, away from zero:

- 4 - 6

-10

2 x \leq -10

ⓘ Both sides are divided by

2

x \leq - \frac{10}{2}

ⓘ We simplify the fraction by dividing the numerator and the denominator by

2

, their greatest common divisor:

- \frac{10}{2}

-5

Solution found:

x \leq -5

▶Solution:

x \in (-∞, -5] \cup [-1, +∞)