x(x+12)>(x-4)^2 — step-by-step solution

1) Let's solve in

ℝ

x \cdot (x + 12) > {(x - 4)}^{2}

ⓘ Expand by multiplying each term inside the parentheses:

x \cdot (x + 12)

x^{2} + x \cdot 12

■Raising a sum to an integer power is multiplying it by itself:

{(x - 4)}^{2}

(x - 4) \cdot (x - 4)

x^{2} + x \cdot 12 > (x - 4) \cdot (x - 4)

ⓘ Expand by multiplying each term inside the parentheses:

(x - 4) \cdot (x - 4)

x \cdot x

+ x \cdot (-4)

- 4 x

- 4 \cdot (-4)

x^{2} + x \cdot 12 > x \cdot x

+ x \cdot (-4)

- 4 x

- 4 \cdot (-4)

ⓘ Multiplying powers with the same base adds the exponents:

x \cdot x

x^{2}

■The minus sign is placed in front of the product:

x \cdot (-4)

- 4 x

■Multiplications are done before additions and subtractions. Two minus signs cancel out, the product of two negative numbers is positive:

-4 \cdot (-4)

16

x^{2} + x \cdot 12 > x^{2}

- 4 x

- 4 x

+ 16

ⓘ Like terms are grouped by adding their coefficients:

- 4 x - 4 x

-8 x

x^{2} + x \cdot 12 > x^{2} - 8 x + 16

x^{2}

+ x \cdot 12

- x^{2}

+ 8 x

- 16 > 0

ⓘ Like terms are grouped by adding their coefficients:

x^{2} + x \cdot 12 - x^{2}

0 + 12 x

12 x + 8 x

20 x

0 + 20 x - 16 > 0

ⓘ Adding 0 changes nothing:

0 + 20 x

20 x

20 x - 16 > 0

ⓘ We add

16

to both sides

20 x > 16

ⓘ Both sides are divided by

20

x > \frac{16}{20}

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

4

, their greatest common divisor:

\frac{16}{20}

\frac{4 \cdot 4}{5 \cdot 4}

\frac{4}{5}

▶Solution obtained:

x > \frac{4}{5}

2) The numeric value of:

x > \frac{4}{5}

equals:

▷

x > 0.8