Given information:

The compound inequality, \(\displaystyle-{2}{\left({x}-{1}\right)}+{4}{<}{x}+{3}\ r \ 5\left(x+2\right)-3\le4x+{r}\ {5}{\left({x}+{2}\right)}-{3}\le{4}{x}+{1} \).

Formula used:

Steps to solve compound inequality:

Step 1: First, solve the compound inequality individually.

Step 2: If two inequalities are combined through “or’/“and” then take the union/intersection, respectively, of individual solution set.

Calculation:

The given compound inequality is \(\displaystyle-{2}{\left({x}-{1}\right)}+{4}{<}{x}+{3}\ r \ 5\left(x+2\right)-3\le4x+{r}\ {5}{\left({x}+{2}\right)}-{3}\le{4}{x}+{1}\). Simplify the inequality.

\(\displaystyle-{2}{\left({x}-{1}\right)}+{4}{<}{x}+{3}\ r \ 5\left(x+2\right)-3\le4x+{r}\ {5}{\left({x}+{2}\right)}-{3}\le{4}{x}+{1}\)

\(\displaystyle-{2}{x}+{2}+{4}{<}{x}+{3}\ r \ 5x+10-3\le4x+{r}\ {5}{x}+{10}-{3}\le{4}{x}+{1}\)

\(\displaystyle-{2}{x}-{x}{<}{3}-{6}\ r \ 5x-4x\le1-{r}\ {5}{x}-{4}{x}\le{1}-{7}\)

\(\displaystyle{x}{>}{1}\ r \ x\le-{r}\ {x}\le-{6}\)

The graph of the solution is:

Therefore, the solution set of compound inequality

\(\displaystyle-{2}{\left({x}-{1}\right)}+{4}{<}{x}+{3}\ r \ 5\left(x+2\right)-3\le4x+{r}\ {5}{\left({x}+{2}\right)}-{3}\le{4}{x}+{1}\) in interval notation is:

\(
(-\infty-6) \cup(1,\infty)\).