(x-1)^2<(x+9)^2 — step-by-step solution

1) Let's solve in

ℝ

{(x - 1)}^{2} < {(x + 9)}^{2}

{(x - 1)}^{2} - {(x + 9)}^{2} < 0

For

{(x - 1)}^{2} - {(x + 9)}^{2}

, we recognize a difference of two squares:

{(x - 1)}^{2} - {(x + 9)}^{2}

. Apply

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

, with

a = x - 1

and

b = x + 9

You can factorize the expression:

{(x - 1)}^{2} - {(x + 9)}^{2}

into:

((x - 1) - (x + 9)) \cdot ((x - 1) + (x + 9))

2) Let's solve in

ℝ

((x - 1) - (x + 9)) \cdot ((x - 1) + (x + 9)) < 0

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

- (x + 9)

- x - 9

■Parentheses are computed first:

- 1 + 9

8

■Like terms are grouped by adding their coefficients:

x + x

2 x

(x

- 1

- x

- 9) \cdot (2 x + 8) < 0

ⓘ Parentheses are computed first:

- 1 - 9

-10

■Like terms are grouped by adding their coefficients:

x - x

0

(0 - 10) \cdot (2 x + 8) < 0

ⓘ Adding 0 changes nothing:

0 - 10

-10

- 10 \cdot (2 x + 8) < 0

We simplify the expression by

-10

and since

-10

is negative, we reverse the inequality sign

⇥Let's solve in

ℝ

2 x + 8 > 0

⇥

2 x > -8

⇥ⓘ Both sides are divided by

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