How do you solve the inequality #6x^2-5x>6#?

1 Answer
Feb 4, 2015

I would start by solving it as a normal equation (2nd degree). So you write:
#6x^2-5x-6>0# (I took the #6# to the left);
Solving your equation you should get two values #x_1=3/2# and #x_2=-2/3#.
Now the tricky bit:
your inequality asks you for values of #x# that make your equation have a value bigger than zero.
They cannot be #x_1 and x_2# because at these point your equation IS equal to zero.
Graphically your function gives the following parabola:

graph{6x^2-5x-6 [-11.96, 13.02, -7.14, 5.34]}

So, basically I have to choose values that are outside the boundaries formed by the two values #x_1 and x_2# to get a value bigger than zero (in the graph the two bits that are ABOVE the #x# axis).!!!
Consider #x_1=3/2=1.5# ok I cannot choose it but what about #2# (which is bigger)?
If I put #x=2# in the equation I get: #6*4-5*2-6=8>0# YES!
Consider now #x_2=-2/3=-0.67# again I cannot choose it but what about #-1#?
If I put #x=-1# in the equation I get: #6*(-1)^2-5*(-1)-6=5>0# YES!
So, OUTSIDE the interval bound by #x_1 and x_2# you can choose #x#.
You express this by writing your solution as:
#-2/3>x>3/2#
Or graphically
enter image source here

Hope it helps