Question

please solve fast with full method​

Answer 1

Answer:

answer

Step-by-step explanation:

answer is = -2,-1/2

Answer 2

$\large\underline{\sf{Given \:Question - }}$

If

$\rm :\longmapsto\: |2x + 1| + |x - 2| + |x + 3| < 8$

then x belongs to

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

$\rm :\longmapsto\: |2x + 1| + |x - 2| + |x + 3| < 8$

The breaking points are - 3, - 1/2 and 2

So,

Following cases arises

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

$\rm :\longmapsto\:2. \: \: - 3 < x < - \dfrac{1}{2}$

$\rm :\longmapsto\:3. \: \: - \dfrac{1}{2} \leqslant x < 2$

$\rm :\longmapsto\:4. \: \: x \geqslant 2$

We know,

$\begin{gathered}\begin{gathered}\bf\: \rm :\longmapsto\: |x| = \begin{cases} &\sf{ - x \: \: when \: x < 0} \\ &\sf{ \: \: \: x \: \: when \: x \geqslant 0} \end{cases}\end{gathered}\end{gathered}$

Consider,

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

The inequality

$\rm :\longmapsto\: |2x + 1| + |x - 2| + |x + 3| < 8$

reduce to

$\rm :\longmapsto\: - (2x + 1) - (x - 2) - (x + 3) < 8$

$\rm :\longmapsto\: - 2x - 1 - x + 2 - x - 3< 8$

$\rm :\longmapsto\: - 4x - 2 < 8$

$\rm :\longmapsto\: - 4x < 8 + 2$

$\rm :\longmapsto\: - 4x < 10$

$\bf\implies \:x > - 2.5$

Hence, there is no solution

$\bf :\longmapsto\:When\: \: - 3 < x < - \: \: \dfrac{1}{2}$

The inequality

$\rm :\longmapsto\: |2x + 1| + |x - 2| + |x + 3| < 8$

reduce to

$\rm :\longmapsto\: - (2x + 1) - (x - 2)+ x + 3 < 8$

$\rm :\longmapsto\: - 2x - 1 - x + 2+ x + 3 < 8$

$\rm :\longmapsto\: - 2x + 4 < 8$

$\rm :\longmapsto\: - 2x < 8 - 4$

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

$\rm :\longmapsto\: x > - 2$

$\bf\implies \:x \: \in \: ( - 2, - \dfrac{1}{2}\bigg)$

Now, Consider

$\bf :\longmapsto\:When\: \: - \dfrac{1}{2} \leqslant x < 2$

The inequality

$\rm :\longmapsto\: |2x + 1| + |x - 2| + |x + 3| < 8$

reduce to

$\rm :\longmapsto\: |2x + 1| + |x - 2| + |x + 3| < 8$

reduce to

$\rm :\longmapsto\: 2x + 1 - (x - 2) + x + 3 < 8$

$\rm :\longmapsto\: 3x + 4 - x + 2 < 8$

$\rm :\longmapsto\: 2x + 6 < 8$

$\rm :\longmapsto\: 2x < 8 - 6$

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

$\bf\implies \:x < 1$

$\bf\implies \:x \: \in \: \bigg( - \: \dfrac{1}{2},1\bigg)$

Now, Consider

$\rm :\longmapsto\:When \: x \geqslant 2$

The inequality

$\rm :\longmapsto\: |2x + 1| + |x - 2| + |x + 3| < 8$

reduce to

$\rm :\longmapsto\: 2x + 1+ x - 2 + x + 3< 8$

$\rm :\longmapsto\: 4x + 2< 8$

$\rm :\longmapsto\: 4x < 8 - 2$

$\rm :\longmapsto\: 4x < 6$

$\bf\implies \:x < \dfrac{3}{2}$

Hence, there is no solution

So, The solution of

$\rm :\longmapsto\: |2x + 1| + |x - 2| + |x + 3| < 8$

is

$\bf\implies \:x \: \in \: \bigg( - 2, - \dfrac{1}{2}\bigg) \: \cup \: \bigg( - \dfrac{1}{2},1\bigg)$

or

$\bf\implies \:x \: \in \: \bigg( - 2,1\bigg) \: - \: \bigg \{ - \dfrac{1}{2}\bigg \}$