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

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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#

Graph.2

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

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.