Question

Solve the following linear inequation and represent the solution set on a number line:
2x − 3 < x +1 ≤ 4x + 7, x ∈ W

Answer 1

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

Given linear inequality is

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

Consider,

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

On Subtracting x, from both sides, we get

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

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

On adding 3 both sides, we get

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

$\bf :\longmapsto\:x < 4 - - - (1)$

Now, Consider

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

On Subtracting 4x, we get

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

$\rm :\longmapsto\: 1 - 3x \leqslant 7$

On Subtracting 1, we get

$\rm :\longmapsto\: 1 - 3x - 1 \leqslant 7 - 1$

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

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

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

$\bf\implies \: - 2 \leqslant x < 4$

As, it is given that

$\rm :\longmapsto\:\boxed{ \tt{ \: x \: \in \: W \: }}$

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

More to know :-

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

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

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

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

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

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

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

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