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
