First, simplify the expression:
#3x^2 − x < 8 − 4x#
#3x^2 +3x - 8 < 0#
Now we can solve for the roots of the "equality" and then check to set the directional inequality limits of the solutions.
#x_1 = 1.208#
#x_2 = -2.208#
#3x^2 − x < 8 − 4x#; #3(1.208^2) − 1.208 < 8 − 4(1.208)#
#4.38 − 1.208 < 8 − 4.832# : #3.172 < 3.168# (incorrect), thus the inequality value must be #x_1 < 1.208#
CHECK:
#3(1.20^2) − 1.20 < 8 − 4(1.20)#
#4.32 − 1.20 < 8 − 4.8# : #3.12 < 3.2# Correct.
AND
#3x^2 − x < 8 − 4x#; #3(-2.208^2) − -2.208 < 8 − 4(-2.208)#
#14.63 + 2.208 < 8 + 8.832# : #16.838 < 16.832# (incorrect), thus the inequality value must be #x_1 > -2.208#
CHECK:
#3(-2.2^2) − -2.2 < 8 − 4(-2.2)#
#14.52 + 2.20 < 8 + 8.8# : #16.72 < 16.8# Correct.
The range is thus:
#-2.208 < x < 1.208#
