Absolute Value Graph help pls?

For reference, my brackets are supposed to be absolute value bars :)

#2 <( 2x-3 ) < 7#

Solve.

2 Answers
May 10, 2018

Break it up into the two smaller inequalities:

#2 < abs(2x - 3)" " and " " abs(2x - 3) < 7#

Our solution will be all #x# that make both inequalities hold.

First inequality:

#abs(2x - 3) > 2<=>{(2x-3 > 2" "or),(2x - 3 < –2):}#

#color(white)(abs(2x - 3) > 2)<=>{(2x > 5" "or),(2x < 1):}#

#color(white)(abs(2x - 3) > 2)<=>{(x > 5/2" "or),(x < 1/2):}#

Second inequality:

#abs(2x-3) < 7<=>{(2x - 3 < 7" "and),(2x - 3 > –7):}#

#color(white)(abs(2x-3) < 7)<=>{(2x < 10" "and),(2x > –4):}#

#color(white)(abs(2x-3) < 7)<=>{(x < 5" "and),(x > –2):}#

Plotting both intervals:

#"<—––——–––—––o   o—————>"#
#"                     o————————o"#
#"<====================>"#
#"    –5 –4 –3 –2 –1  0   1   2   3   4   5   6"#

We can see they overlap on #(–2, 1/2) uu (5/2, 5)#. This is our solution:

#x in (–2, 1/2) uu (5/2, 5)#.

Below is a graph of #y = abs(2x - 3)#, marked with horizontal lines at 2 and 7. In order for #y# to be between 2 and 7, we must choose an #x# in #(–2," " 0.5)# or #(2.5," " 5)#.

graph{(y - abs(2x - 3))(y - 2)(y - 7) = 0 [-10, 10, -1, 9]}

May 10, 2018

Quicker method.

Explanation:

#2 < abs(2x - 3) < 7#

#<=>{(2 < +(2x - 3) < 7" "or),(2 < -(2x-3) < 7):}#

#<=>{(2 <"   " 2x - 3 < 7" "or),(2 < –2x+3 < 7):}#

#<=>{("   "5 < "   "2x < 10" "or),(–1 < –2x < "  "4):}#

#<=>{(5/2 < x < "   "5" "or),(1/2 > x > –2):}#

Solution set: #x in (–2," " 1/2) uu (5/2, " "5)#