Work out the integer values that satisfy: x^2−7 x+ 11<0

It's easier if we first visualise the function on a graph, then we can see in which regions it is negative. From there we can see which integer values lie in this region. A positive quadratic function has the shape of a U, and remember a negative quadratic function has the shape of an n. To sketch the graph you will want to find the roots of the function, to see where the curve crosses the x axis. Solve f(x)=0. Does it factorise? no. Use quadratic formula: x = (7+-sqrt(49-44))/2 . Sketch the U curve making it cross the x axis at these 2 values. (go to whiteboard) Observe for what values of x are the y values negative (curve below the x axis). You should find that between the two roots the graph is negative ( the bottom of the U). Therefore the integer values we are looking for are those lying between 2.38 and 4.61, i.e. 3,4.
You should now check the values 2,3,4,5 and check that 2 and 5 are positive, while 3 and 4 are negative. An alternative is that when you know roughly what the roots are, you can start guessing integers that might be positive and negative and work it out that way. After doing a few of these questions, you can speed up the process: solve for roots. what are the integers between the roots? But make sure you understand why this works.