Question

Solve the inequation:
2x-3 < x+1 ≤ 4x+7 , if x is integer.

Answer 1

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

Given inequality is

$\rm :\longmapsto\:2x - 3 < x + 1 \leqslant 4x + 7$

Consider,

$\rm :\longmapsto\:2x - 3 < x + 1$

$\rm :\longmapsto\:2x - x < 3 + 1$

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

$\bf\implies \:x \: \in \: ( - \infty ,4) - - - (1)$

Now, Consider

$\rm :\longmapsto\: x + 1 \leqslant 4x + 7$

$\rm :\longmapsto\: x - 4x \leqslant 7 - 1$

$\rm :\longmapsto\: - 3x \leqslant 6$

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

$\bf\implies \:x \: \in \: [ - 2, \infty ) - - - (2)$

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

$\bf\implies \:x \: \in \: [ - 2,4)$

It is given that

$\bf :\longmapsto\:x \: \in \: Z$

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

Additional Information :-

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

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

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

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

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

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

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

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