|-3x+2|=x+6 — step-by-step solution

1) Let's solve in

ℝ

| - 3 x + 2 | = x + 6

The absolute-value arguments are zero at

\frac{2}{3}

. Split ℝ at these points and solve on each interval

(-∞, \frac{2}{3}]

- 3 x + 2 \geq 0

, so

| - 3 x + 2 | = - 3 x + 2

(-∞, \frac{2}{3}]

, the equation becomes

- 3 x + 2 = x + 6

2) Let's solve in

ℝ

- 3 x + 2 = x + 6

ⓘ We gather the terms in

x

on the right and the other terms on the left

2 - 6 = x + 3 x

ⓘ Start from

2

and move back

6

on the number line, going below zero:

2 - 6

-4

■Like terms are grouped by adding their coefficients:

x + 3 x

4 x

-4 = 4 x

ⓘ Both sides are divided by

4

x = - \frac{4}{4}

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

4

, their greatest common divisor:

- \frac{4}{4}

-1

Solution obtained:

x = -1

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

, we can keep the solution

-1

[\frac{2}{3}, +∞)

- 3 x + 2 \leq 0

, so

| - 3 x + 2 | = - (- 3 x + 2)

[\frac{2}{3}, +∞)

, the equation becomes

- (- 3 x + 2) = x + 6

3) Let's solve in

ℝ

- (- 3 x + 2) = x + 6

ⓘ A minus sign before a parenthesis changes the sign of each of its terms:

- (- 3 x + 2)

3 x - 2

3 x - 2 = x + 6

ⓘ We add

2

to both sides■Start from

6

and move forward

2

on the number line:

6 + 2

8

3 x = x + 8

ⓘ We subtract

x

from both sides

3 x - x = 8

ⓘ Like terms are grouped by adding their coefficients:

3 x - x

2 x

2 x = 8

ⓘ Both sides are divided by

2

x = \frac{8}{2}

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

2

, their greatest common divisor:

\frac{8}{2}

4

Solution found:

x = 4

4 \geq \frac{2}{3}

, we can keep the solution

4

▶Solutions found:

x = {-1, 4}