How do you solve for x in #3x-5 < x + 9 \le 5x + 13 #?

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Answer 1 · 2014-10-31T15:36:56

By separating into two inequalities,

#{(3x-5 < x+9),(x+9 le 5x+13):}#

Let us work on the first inequality.

#3x-5 < x+9#

by subtracting #x#,

#=> 2x-5 < 9#

by adding 5,

#=> 2x<14#

by dividing by #2#,

#=> x<7#

Let us work on the second inequality.

#x+9 le 5x+13#

by subtracting #x#,

#=> 9 le 4x+13#

by subtracting #13#,

#=> -4 le 4x#

by dividing by #4#,

#=> -1 le x#

By combining the two inequalities, we have

#-1 le x < 7#,

or in interval notation, the solution set is

#[-1,7)#.

I hope that this was helpful.