Question

23. Solve the following inequation, write down the solution set and represent it on the real
number line:
-2 + 10x < 13x + 10 < 24 + 10x, x + Z.
(2018)​

Answer 1

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

Given linear inequality is

$\rm :\longmapsto\: - 2 + 10x < 13x + 10 < 24 + 10x \: \forall \: x \: \in \: Z$

Consider,

$\rm :\longmapsto\: - 2 + 10x < 13x + 10$

On adding 2 on both sides, we get

$\rm :\longmapsto\: - 2 + 10x + 2 < 13x + 10 + 2$

$\rm :\longmapsto\: 10x < 13x + 12$

On Subtracting 13x from both sides, we get

$\rm :\longmapsto\: 10x - 13x < 13x + 12 - 13x$

$\rm :\longmapsto\: -3x < 12$

$\rm :\implies\:x > - 4 - - - (1)$

Now,

Consider,

$\rm :\longmapsto\: 13x + 10 < 24 + 10x \:$

Subtracting 10x from both sides, we get

$\rm :\longmapsto\: 13x + 10 - 10x< 24 + 10x - 10x \:$

$\rm :\longmapsto\: 3x + 10 < 24$

On Subtracting 10 from both sides, we get

$\rm :\longmapsto\: 3x + 10 - 10< 24 - 1 0$

$\rm :\longmapsto\: 3x < 14$

$\rm :\implies\:x < \dfrac{14}{3} - - - (2)$

From equation (1) and (2), we concluded that

$\bf\implies \: - 4 < x < \dfrac{14}{3}$

As x is an integer. so x can take values

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

Additional Information :-

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

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

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

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

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

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

$\boxed{ \sf{ \: \frac{x}{ \: \: y \: \: } > 0 \implies \: x > 0,y > 0 \: \: or \: \: x < 0,y < 0}}$

$\boxed{ \sf{ \: \frac{x}{ \: \: y \: \: } < 0 \implies \: x > 0,y < 0 \: \: or \: \: x < 0,y > 0}}$