91Ó°ÊÓ

The given inequality is: $-2<x<4$.

The given inequality $-2<x<4$ implies that $x$ is greater than $-2$ but less than 4.

Here inequality is not having equal to sign, therefore the values $-2$ and 4 are excluded.

The graph of the given inequality $-2<x<4$ on the number line is:

The given inequality is: $x<-1orx>3$.

The given inequality $x<-1orx>3$ implies that $x$ is less than $-1$ and also $x$ is greater than 3.

Here inequality is not having equal to sign, therefore the values $-1$ and 3 are excluded.

The graph of the given inequality$x<-1orx>3$ on the number line is:

The given inequality is: $\left(-2<x<4\right)and\left(x<-1orx>3\right)$.

The solution of the given inequality $\left(-2<x<4\right)and\left(x<-1orx>3\right)$ will be the intersection of the graphs of the inequalities $-2<x<4$ and $x<-1orx>3$.

The graph of the given inequality $\left(-2<x<4\right)and\left(x<-1orx>3\right)$ on the number line is:

The given inequality is: $3<\left|x+2\right|≤8$.

The solution of the given inequality $3<\left|x+2\right|≤8$can be find out by finding the intersection of the solutions of the inequalities $\left|x+2\right|≤8$ and $\left|x+2\right|>3$.

As, it is known that if $\left|a\right|≤b$, then $-b≤a≤b$.

Therefore, the solution of the inequality $\left|x+2\right|≤8$is given by:

$\begin{array}{c}\mathbf{−}\mathbf{8}\mathbf{≤}\mathbf{x}\mathbf{+}\mathbf{2}\mathbf{≤}\mathbf{8}\\ \mathbf{−}\mathbf{8}\mathbf{−}\mathbf{2}\mathbf{≤}\mathbf{x}\mathbf{+}\mathbf{2}\mathbf{−}\mathbf{2}\mathbf{≤}\mathbf{8}\mathbf{−}\mathbf{2}\\ \mathbf{−}\mathbf{10}\mathbf{≤}\mathbf{x}\mathbf{≤}\mathbf{6}\end{array}$

Therefore, the solution of the inequality $\left|x+2\right|≤8$is $-10≤x≤6$.

As, it is known that if $\left|a\right|>b$, then $a>b\text{or a}<-b$.

Therefore, the solution of the inequality $\left|x+2\right|>3$is given by:

$\begin{array}{c}x+2>3\text{or}x+2<−3\\ x+2−2>3−2\text{or}x+2−2<−3−2\\ x>1\text{or}x<−5\end{array}$

Therefore, the solution of the inequality $\left|x+2\right|>3$is $x>1\text{or}x<-5$.

As, the solution of the given inequality $3<\left|x+2\right|≤8$can be find out by finding the intersection of the solutions of the inequalities $\left|x+2\right|≤8$ and $\left|x+2\right|>3$.

Therefore, the solution of the given inequality $3<\left|x+2\right|≤8$is given by:

$\left(-10≤x≤6\right)âˆ\copyright \left(x>1\text{or}x<-5\right)=-10≤x<-5\text{or 1}<x≤6$

Therefore, the solution of the given inequality $3<\left|x+2\right|≤8$is $-10≤x<-5\text{or 1}<x≤6$.

The graph of the given inequality$-10≤x<-5\text{or 1}<x≤6$ on the number line is: