|4-x|=|2x+1| — step-by-step solution

1) Let's solve in

ℝ

| 4 - x | = | 2 x + 1 |

Both sides are nonnegative. Comparing them is equivalent to comparing their squares, with the same comparison sign

2) Let's solve in

ℝ

{(4 - x)}^{2} = {(2 x + 1)}^{2}

{(4 - x)}^{2} - {(2 x + 1)}^{2} = 0

For

{(4 - x)}^{2} - {(2 x + 1)}^{2}

, we recognize a difference of two squares:

{(4 - x)}^{2} - {(2 x + 1)}^{2}

. Apply

a^{2} - b^{2} = (a - b) \cdot (a + b)

, with

a = 4 - x

and

b = 2 x + 1

You can write the expression:

{(4 - x)}^{2} - {(2 x + 1)}^{2}

in factored form:

((4 - x) - (2 x + 1)) \cdot ((4 - x) + (2 x + 1))

3) Let's solve in

ℝ

((4 - x) - (2 x + 1)) \cdot ((4 - x) + (2 x + 1)) = 0

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

- (2 x + 1)

- 2 x - 1

■Parentheses are computed first:

4 + 1

5

■Like terms are grouped by adding their coefficients:

- x + 2 x

x

(4

- x

- 2 x

- 1) \cdot (5 + x) = 0

ⓘ Parentheses are computed first:

4 - 1

3

■Like terms are grouped by adding their coefficients:

- x - 2 x

-3 x

(3 - 3 x) \cdot (5 + x) = 0

A product is zero if and only if one of its factors is zero: we solve each factor equal to 0

⇥3.1) Let's solve in

ℝ

3 - 3 x = 0

⇥

3 x = 3

⇥ⓘ 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 obtained:

x = 1

⇥3.2) Let's solve in

ℝ

5 + x = 0

⇥Solution obtained:

x = -5

▶Solution:

x = {-5, 1}