|x-3|=|3x+5| — step-by-step solution

1) Let's solve in

ℝ

| x - 3 | = | 3 x + 5 |

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

2) Let's solve in

ℝ

{(x - 3)}^{2} = {(3 x + 5)}^{2}

{(x - 3)}^{2} - {(3 x + 5)}^{2} = 0

For

{(x - 3)}^{2} - {(3 x + 5)}^{2}

, we recognize a difference of two squares:

{(x - 3)}^{2} - {(3 x + 5)}^{2}

. Apply

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

, with

a = x - 3

and

b = 3 x + 5

You can factor:

{(x - 3)}^{2} - {(3 x + 5)}^{2}

into:

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

3) Let's solve in

ℝ

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

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

- (3 x + 5)

- 3 x - 5

■Parentheses are computed first:

- 3 + 5

2

■Like terms are grouped by adding their coefficients:

x + 3 x

4 x

(x

- 3

- 3 x

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

ⓘ Parentheses are computed first:

- 3 - 5

-8

■Like terms are grouped by adding their coefficients:

x - 3 x

-2 x

(- 2 x - 8) \cdot (4 x + 2) = 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

- 2 x - 8 = 0

⇥

- 2 x = 8

⇥ⓘ Both sides are divided by

-2

⇥

x = \frac{8}{-2}

⇥ⓘ The sign of a fraction goes in front of the bar, whether it comes from the numerator or the denominator:

\frac{8}{-2}

- \frac{8}{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

⇥3.2) Let's solve in

ℝ

4 x + 2 = 0

⇥

4 x = -2

⇥ⓘ Both sides are divided by

4

⇥

x = - \frac{2}{4}

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

2

, their greatest common divisor:

- \frac{2}{4}

- \frac{1 \cdot 2}{2 \cdot 2}

- \frac{1}{2}

⇥Solution obtained:

x = - \frac{1}{2}

▶Solution:

x = {-4, - \frac{1}{2}}

4) The numeric value of:

x = {-4, - \frac{1}{2}}

equals:

▷

x = {-4, -0.5}