To calculate: The solution of the compound inequality -2\left(x-1\right)+4<x+3\ \text

mronjo7n 2021-11-22 Answered

To calculate: The solution of the compound inequality
\(\displaystyle-{2}{\left({x}-{1}\right)}+{4}{<}{x}+{3}\ r \ 5\left(x+2\right)-3\le4x+{r}\ {5}{\left({x}+{2}\right)}-{3}\le{4}{x}+{1}\)
and graph the solution set and write the solution set in interval notation.

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Expert Answer

jackbranv5
Answered 2021-11-23 Author has 159 answers

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

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Thomas Conway
Answered 2021-11-24 Author has 405 answers

Given information:
The compound inequality, \(-2\left(x-1\right)+4<x+3\ \text or \ 5\left(x+2\right)-3\le4x+1\).
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 \(-2\left(x-1\right)+4<x+3\ \text or \ 5\left(x+2\right)-3\le4x+1\). Simplify the inequality.
\(-2\left(x-1\right)+4<x+3\ \text or \ 5\left(x+2\right)-3\le4x+1\)
\(-2x+2+4<x+3 \ \text or \ 5x+10-3\le4x+1\)
\(-2x-x<3-6 \ \text or \ 5x-4x\le1-7\)
\(x>1 \ \text or \ x\le-6\)
The graph of the solution is:
image
Therefore, the solution set of compound inequality \(-2\left(x-1\right)+4<x+3\ \text or \ 5\left(x+2\right)-3\le4x+1\) in interval notation is: \(\left(-\propto-6\mid\cup\left(1,\propto\right)\right)\).
Given information:
 

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