# How do you solve and graph the compound inequality x- 3 > 3 and -x + 1 < -2 ?

Jul 20, 2018

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#### Explanation:

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We are given the Compound Inequality:

color(red)((x-3) > 3 " AND " (-x+1)<(-2)

We can solve these inequalities separately.

Since $\textcolor{red}{\text{ AND }}$ is used to Join the two inequalities,
the final result will be the intersection of the two given inequalities.

Inequality-1

$\left(x - 3\right) > 3$

Add color(red)(3 to both sides of the inequality.

$\Rightarrow \left(x - 3\right) + 3 > 3 + 3$

$\Rightarrow x - \cancel{3} + \cancel{3} > 3 + 3$

$\Rightarrow x > 6$ ...Res.1

Inequality-2

$\left(- x + 1\right) < \left(- 2\right)$

Subtract color(red)((-1) from both sides of the inequality.

$\Rightarrow \left(- x + 1\right) - 1 < \left(- 2\right) - 1$

$\Rightarrow - x + \cancel{1} - \cancel{1} < - 2 - 1$

$\Rightarrow - x < - 3$

Multiply both sides of the inequality by color(red)((-1) and please remember to reverse the inequality:

$\Rightarrow \left(- x\right) \left(- 1\right) > \left(- 3\right) \left(- 1\right)$

$\Rightarrow x > 3$ ... Res.2

Using the intermediate results (Res.1) and (Res.2), we get

color(blue)(x>6 " AND " x>3

FINAL SOLUTION:

color(red)(x>6

Using Interval Notation:

color(red)((6,oo)

Important Note:

Dotted Lines in all the graphs below represent a solution that does not include a certain value, indicated by the dotted line.

Graph.1

Graph of the inequality: $\left(x - 3\right) > 3$

Graph.2

Graph of the inequality: $\left(- x + 1\right) < \left(- 2\right)$

Graph.3: Solution Graph

color(red)((x-3) > 3 " AND " (-x+1)<(-2)

Overlapping area in the graph is our required solution,

Compare this graph with the following graph.

Both the graphs represent the same solution.

Graph.4: Solution Graph color(red)(x>6

Hope this helps.