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

1 Answer
Jul 20, 2018

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Please read the explanation.

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 color(red)(" AND ") is used to Join the two inequalities,
the final result will be the intersection of the two given inequalities.

Inequality-1

(x-3) > 3

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

rArr (x-3)+3>3+3

rArr x-cancel 3+cancel 3>3+3

rArr x>6 ...Res.1

Inequality-2

(-x+1)<(-2)

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

rArr (-x+1)-1<(-2)- 1

rArr -x+ cancel 1- cancel 1<-2- 1

rArr -x< -3

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

rArr (-x)(-1)>(-3)(-1)

rArr 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: (x-3) > 3

enter image source here

Graph.2

Graph of the inequality: (-x+1)<(-2)

enter image source here

Graph.3: Solution Graph

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

enter image source here

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

enter image source here

Hope this helps.