Question

INECUASION

21x²>x+12

Answer 1

$\huge\tt{Answer:-}$

$\sf \color{fuchsia} x > +\dfrac{\sqrt{1009}}{42} + \dfrac{1}{42}$

$\sf \color{fuchsia} or$

$\sf \color{fuchsia} x > -\dfrac{\sqrt{1009}}{42} + \dfrac{1}{42}$

⠀ ⠀⠀ ⠀ ⠀⠀ ⠀

$\huge\tt{Solution:-}$

$\sf 21x² > x + 12$

⠀⠀ ⠀ ⠀⠀ ⠀

$\sf By \: dividing \: both \: sides \: by \: 21$

$\sf \dfrac{21x²}{21} > \dfrac{x+12}{21}$

$\sf \dfrac{\cancel{21}x²}{\cancel{21}} > \dfrac{x}{21} + \dfrac{12}{21}$

$\sf x² > \dfrac{x}{21} + \dfrac{\cancel{12}^{4}}{\cancel{21}_{7}}$

$\sf x² > \dfrac{x}{21} + \dfrac{4}{7}$

$\sf x² - \dfrac{x}{21} > \dfrac{4}{7}$

⠀⠀ ⠀ ⠀⠀ ⠀

$\sf By \: adding \: (\dfrac{-1}{2} × coefficient \: of \: x)² \: to \: both \: sides$

$\sf Here \: (\dfrac{-1}{2} × coefficient \: of \: x)² = (\dfrac{-1}{2} × \dfrac{-1}{21})²$

$\sf = (\dfrac{1}{42})²$

⠀ ⠀⠀ ⠀ ⠀⠀ ⠀

$\sf \implies x² - \dfrac{x}{21} + (\dfrac{1}{42})² > \dfrac{4}{7} + (\dfrac{1}{42})²$

$\sf \implies x² - 2 × x × \dfrac{1}{42} + (\dfrac{1}{42})² > \dfrac{4}{7} + \dfrac{1}{1764}$

$\sf \implies (x - \dfrac{1}{42})² > \dfrac{1008}{1764} + \dfrac{1}{1764}$

$\sf \implies (x - \dfrac{1}{42})² > \dfrac{1008+1}{1764}$

$\sf \implies (x - \dfrac{1}{42})² > \dfrac{1009}{1764}$

$\sf \implies (x - \dfrac{1}{42}) > ±\sqrt{\dfrac{1009}{1764}}$

$\sf \implies x - \dfrac{1}{42} > ±\sqrt{\dfrac{1009}{(42)²}}$

$\sf \implies x - \dfrac{1}{42} > ±\dfrac{\sqrt{1009}}{42}$

$\sf \implies x > ±\dfrac{\sqrt{1009}}{42} + \dfrac{1}{42}$

⠀ ⠀⠀ ⠀ ⠀⠀ ⠀

$\boxed{\sf x > ±\dfrac{\sqrt{1009}}{42} + \dfrac{1}{42}}$

⠀⠀ ⠀ ⠀⠀ ⠀

$\sf \color{fuchsia} Therefore, \: x > +\dfrac{\sqrt{1009}}{42} + \dfrac{1}{42}$

$\sf \color{fuchsia} or$

$\sf \color{fuchsia} x > -\dfrac{\sqrt{1009}}{42} + \dfrac{1}{42}$

Answer 2

Step-by-step explanation:

\huge\tt{Answer:-}Answer:−

\sf \color{fuchsia} x > +\dfrac{\sqrt{1009}}{42} + \dfrac{1}{42}x>+

42

1009

+

42

1

\sf \color{fuchsia} oror

\sf \color{fuchsia} x > -\dfrac{\sqrt{1009}}{42} + \dfrac{1}{42}x>−

42

1009

+

42

1

⠀ ⠀⠀ ⠀ ⠀⠀ ⠀

\huge\tt{Solution:-}Solution:−

\sf 21x² > x + 1221x²>x+12

⠀⠀ ⠀ ⠀⠀ ⠀

\sf By \: dividing \: both \: sides \: by \: 21Bydividingbothsidesby21

\sf \dfrac{21x²}{21} > \dfrac{x+12}{21}

21

21x²

>

21

x+12

\sf \dfrac{\cancel{21}x²}{\cancel{21}} > \dfrac{x}{21} + \dfrac{12}{21}

21

21

x²

>

21

x

+

21

12

\sf x² > \dfrac{x}{21} + \dfrac{\cancel{12}^{4}}{\cancel{21}_{7}}x²>

21

x

+

21

7

12

4

\sf x² > \dfrac{x}{21} + \dfrac{4}{7}x²>

21

x

+

7

4

\sf x² - \dfrac{x}{21} > \dfrac{4}{7}x²−

21

x

>

7

4

⠀⠀ ⠀ ⠀⠀ ⠀

\sf By \: adding \: (\dfrac{-1}{2} × coefficient \: of \: x)² \: to \: both \: sidesByadding(

2

−1

×coefficientofx)²tobothsides

\sf Here \: (\dfrac{-1}{2} × coefficient \: of \: x)² = (\dfrac{-1}{2} × \dfrac{-1}{21})²Here(

2

−1

×coefficientofx)²=(

2

−1

×

21

−1

)²

\sf = (\dfrac{1}{42})²=(

42

1

)²

⠀ ⠀⠀ ⠀ ⠀⠀ ⠀

\sf \implies x² - \dfrac{x}{21} + (\dfrac{1}{42})² > \dfrac{4}{7} + (\dfrac{1}{42})²⟹x²−

21

x

+(

42

1

)²>

7

4

+(

42

1

)²

\sf \implies x² - 2 × x × \dfrac{1}{42} + (\dfrac{1}{42})² > \dfrac{4}{7} + \dfrac{1}{1764}⟹x²−2×x×

42

1

+(

42

1

)²>

7

4

+

1764

1

\sf \implies (x - \dfrac{1}{42})² > \dfrac{1008}{1764} + \dfrac{1}{1764}⟹(x−

42

1

)²>

1764

1008

+

1764

1

\sf \implies (x - \dfrac{1}{42})² > \dfrac{1008+1}{1764}⟹(x−

42

1

)²>

1764

1008+1

\sf \implies (x - \dfrac{1}{42})² > \dfrac{1009}{1764}⟹(x−

42

1

)²>

1764

1009

\sf \implies (x - \dfrac{1}{42}) > ±\sqrt{\dfrac{1009}{1764}}⟹(x−

42

1

)>±

1764

1009

\sf \implies x - \dfrac{1}{42} > ±\sqrt{\dfrac{1009}{(42)²}}⟹x−

42

1

>±

(42)²

1009

\sf \implies x - \dfrac{1}{42} > ±\dfrac{\sqrt{1009}}{42}⟹x−

42

1

>±

42

1009

\sf \implies x > ±\dfrac{\sqrt{1009}}{42} + \dfrac{1}{42}⟹x>±

42

1009

+

42

1

⠀ ⠀⠀ ⠀ ⠀⠀ ⠀

\boxed{\sf x > ±\dfrac{\sqrt{1009}}{42} + \dfrac{1}{42}}

x>±

42

1009

+

42

1

⠀⠀ ⠀ ⠀⠀ ⠀

\sf \color{fuchsia} Therefore, \: x > +\dfrac{\sqrt{1009}}{42} + \dfrac{1}{42}Therefore,x>+

42

1009

+

42

1

\sf \color{fuchsia} oror

\sf \color{fuchsia} x > -\dfrac{\sqrt{1009}}{42} + \dfrac{1}{42}x>−

42

1009

+

42

1