|4-x|=2x+1 — step-by-step solution

1) Let's solve in

ℝ

| 4 - x | = 2 x + 1

The absolute-value arguments are zero at

4

. Split ℝ at these points and solve on each interval

(-∞, 4]

4 - x \geq 0

, so

| 4 - x | = 4 - x

(-∞, 4]

, the equation becomes

4 - x = 2 x + 1

2) Let's solve in

ℝ

4 - x = 2 x + 1

ⓘ We gather the terms in

x

on the right and the other terms on the left

4 - 1 = 2 x + x

ⓘ Start from

4

and move back

1

on the number line:

4 - 1

3

■Like terms are grouped by adding their coefficients:

2 x + x

3 x

3 = 3 x

ⓘ Both sides are divided by

3

x = \frac{3}{3}

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

3

, their greatest common divisor:

\frac{3}{3}

1

Solution found:

x = 1

1 \leq 4

, we can keep the solution

1

[4, +∞)

4 - x \leq 0

, so

| 4 - x | = - (4 - x)

[4, +∞)

, the equation becomes

- (4 - x) = 2 x + 1

3) Let's solve in

ℝ

- (4 - x) = 2 x + 1

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

- (4 - x)

- 4 + x

- 4 + x = 2 x + 1

ⓘ We gather the terms in

x

on the right and the other terms on the left

- 4 - 1 = 2 x - x

ⓘ Start from

-4

and move back

1

more on the number line, away from zero:

- 4 - 1

-5

■Like terms are grouped by adding their coefficients:

2 x - x

x

-5 = x

Solution found:

x = -5

-5 \notin [4, +∞)

, we cannot keep the solution

-5

▶Solution found:

x = 1