|7x+7|=|4x-19| — step-by-step solution

1) Let's solve in

ℝ

| 7 x + 7 | = | 4 x - 19 |

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

2) Let's solve in

ℝ

{(7 x + 7)}^{2} = {(4 x - 19)}^{2}

{(7 x + 7)}^{2} - {(4 x - 19)}^{2} = 0

For

{(7 x + 7)}^{2} - {(4 x - 19)}^{2}

, we recognize a difference of two squares:

{(7 x + 7)}^{2} - {(4 x - 19)}^{2}

. Apply

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

, with

a = 7 x + 7

and

b = 4 x - 19

You can factor:

{(7 x + 7)}^{2} - {(4 x - 19)}^{2}

into:

((7 x + 7) - (4 x - 19)) \cdot ((7 x + 7) + (4 x - 19))

3) Let's solve in

ℝ

((7 x + 7) - (4 x - 19)) \cdot ((7 x + 7) + (4 x - 19)) = 0

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

- (4 x - 19)

- 4 x + 19

■Parentheses are computed first:

7 - 19

-12

■Like terms are grouped by adding their coefficients:

7 x + 4 x

11 x

(7 x

+ 7

- 4 x

+ 19) \cdot (11 x - 12) = 0

ⓘ Parentheses are computed first:

7 + 19

26

■Like terms are grouped by adding their coefficients:

7 x - 4 x

3 x

(3 x + 26) \cdot (11 x - 12) = 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 x + 26 = 0

⇥

3 x = -26

⇥ⓘ Both sides are divided by

3

⇥Solution obtained:

x = - \frac{26}{3}

⇥3.2) Let's solve in

ℝ

11 x - 12 = 0

⇥

11 x = 12

⇥ⓘ Both sides are divided by

11

⇥Solution found:

x = \frac{12}{11}

▶Solution:

x = {- \frac{26}{3}, \frac{12}{11}}

4) The numeric value of:

x = {- \frac{26}{3}, \frac{12}{11}}

equals:

▷

x = {\approx -8.6667, \approx 1.0909}