Solve: m(m-3) < 54

m^2-3m-54 < 0
Since the leading term is positive the parabola opens upwards. After factoring...
(m+6)(m-9) < 0

Using the vertex formula we see that min is at -b/2a=3/2 we know the expression is negative between -6 and 9...
Solution interval:
(-6,9)

solve x + 1/x < (1/2)(7-x)

this one is quite long - I'll outline it for you :) 


 

eventually leads to 
 

which leads us to test about the points x=0, x=1/3, and x=2 

after testing on these intervals we find 

 
or  

:)

How would one go about solving the following inequality in interval notation? 2x^2+x-8>0

Since this expression is quadratic and it's leading coefficient is positive, it must open upwards. Therefore if it has real roots, then the quadratic expression will be positive for x-values greater than those roots. In fact it has the roots: 
 
which is about...
(-oo,-2.26) U (1.76,oo) 

Solve the inequality and express the solution in interval form.