|4x-5|=|9-4x| — step-by-step solution

1) Let's solve in

ℝ

| 4 x - 5 | = | 9 - 4 x |

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 - 5)}^{2} = {(9 - 4 x)}^{2}

{(4 x - 5)}^{2} - {(9 - 4 x)}^{2} = 0

For

{(4 x - 5)}^{2} - {(9 - 4 x)}^{2}

, we recognize a difference of two squares:

{(4 x - 5)}^{2} - {(9 - 4 x)}^{2}

. Apply

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

, with

a = 4 x - 5

and

b = 9 - 4 x

You can factor:

{(4 x - 5)}^{2} - {(9 - 4 x)}^{2}

into:

((4 x - 5) - (9 - 4 x)) \cdot ((4 x - 5) + (9 - 4 x))

3) Let's solve in

ℝ

((4 x - 5) - (9 - 4 x)) \cdot ((4 x - 5) + (9 - 4 x)) = 0

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

- (9 - 4 x)

- 9 + 4 x

■Parentheses are computed first:

- 5 + 9

4

■Like terms are grouped by adding their coefficients:

4 x - 4 x

0

(4 x

- 5

- 9

+ 4 x) \cdot (0 + 4) = 0

ⓘ Parentheses are computed first:

- 5 - 9

-14

■Like terms are grouped by adding their coefficients:

4 x + 4 x

8 x

■Adding 0 changes nothing:

0 + 4

4

(8 x - 14) \cdot 4 = 0

We simplify the expression by

4

⇥Let's solve in

ℝ

8 x - 14 = 0

⇥

8 x = 14

⇥ⓘ Both sides are divided by

8

⇥

x = \frac{14}{8}

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

2

, their greatest common divisor:

\frac{14}{8}

\frac{7 \cdot 2}{4 \cdot 2}

\frac{7}{4}

▶Solution found:

x = \frac{7}{4}

4) The numeric value of:

x = \frac{7}{4}

equals:

▷

x = 1.75