Question

Solve the following inequation and write the solution set :

11x - 4 < 15x + 4 < 13x + 14, x € W.
​

Answer 1

$\large\underline{\sf{Solution-}}$

↝ Given that

$\rm :\longmapsto\:11x - 4 < 15x + 4 < 13x + 14$

where x belongs to W.

Consider,

$\rm :\longmapsto\:11x - 4 < 15x + 4$

On Subtracting 15x from both sides, we get

$\rm :\longmapsto\:11x - 4 - 15x < 15x + 4 - 15x$

$\rm :\longmapsto\: - 4x - 4 < 4$

On adding 4 both sides, we get

$\rm :\longmapsto\: - 4x - 4 + 4< 4 + 4$

$\rm :\longmapsto\: - 4x < 8$

$\bf\implies \:x > - 2$

As x belongs to whole numbers,

$\bf\implies \:x \: = \: \{0, \: 1, \: 2, \: 3, \:. \:. \: . \: \} - - - (1)$

Consider,

$\rm :\longmapsto\:15x + 4 < 13x + 14$

On Subtracting 13x from both sides, we get

$\rm :\longmapsto\:15x + 4 - 13x < 13x + 14 - 13x$

$\rm :\longmapsto\:2x + 4 < 14$

On Subtracting 4 both sides, we get

$\rm :\longmapsto\:2x + 4 - 4 < 14 - 4$

$\rm :\longmapsto\:2x < 10$

$\bf\implies \:x < 5$

As x belongs to whole numbers,

$\bf\implies \:x \: = \: \{0, \: 1, \: 2, \: 3, \: 4 \} - - - (2)$

So, from (1) and (2),

$\bf\implies \:x \: = \: \{0, \: 1, \: 2, \: 3, \: 4 \} \: \cap \: \{0, \: 1, \: 2, \: 3,...\}$

$\bf\implies \:x \: = \: \{0, \: 1, \: 2, \: 3, \: 4 \}$

Additional Information :-

$\green{ \boxed{ \bf \: x > y \: \implies \: - x < - y}}$

$\green{ \boxed{ \bf \: x < y \: \implies \: - x > - y}}$

$\green{ \boxed{ \bf \: x \geqslant y \: \implies \: - x \leqslant - y}}$

$\green{ \boxed{ \bf \: x \leqslant y \: \implies \: - x \geqslant - y}}$

$\green{ \boxed{ \bf \: x > - \: y \: \implies \: - x < y}}$

Answer 2

Solution−

↝ Given that

:⟼11x−4<15x+4<13x+14

where x belongs to W.

Consider,

:⟼11x−4<15x+4

On Subtracting 15x from both sides, we get

:⟼11x−4−15x<15x+4−15x

:⟼−4x−4<4

On adding 4 both sides, we get

⟼−4x−4+4<4+4

⟼−4x<8

⟹x>−2

As x belongs to whole numbers,

⟹x={0,1,2,3,...}−−−(1)

Consider,

:⟼15x+4<13x+14

On Subtracting 13x from both sides, we get

:⟼15x+4−13x<13x+14−13x

:⟼2x+4<14

On Subtracting 4 both sides, we get

:⟼2x+4−4<14−4

⟼2x<10

⟹x<5

As x belongs to whole numbers,

⟹x={0,1,2,3,4}−−−(2)

So, from (1) and (2),

⟹x={0,1,2,3,4}∩{0,1,2,3,...}

⟹x={0,1,2,3,4}

Additional Information :-

$\green{ \boxed{ \bf \: x > y \: \implies \: - x < - y}}$

$</p><p>\green{ \boxed{ \bf \: x < y \: \implies \: - x > - y}}$

$</p><p>\green{ \boxed{ \bf \: x \geqslant y \: \implies \: - x \leqslant - y}}$

$</p><p>\green{ \boxed{ \bf \: x \leqslant y \: \implies \: - x \geqslant - y}}$

$\green{ \boxed{ \bf \: x > - \: y \: \implies \: - x < y}} </p><p>$