Question

Find the set of values of x for which

a.  

b.  

c.  both  and .

Solution

a.
 

We are given;

b.
 

We are required to solve the inequality;

We solve the following equation to find critical values of ;

Now we have two options;

Hence the critical points on the curve for the given condition are -3 & 12.

Standard form of quadratic equation is;

The graph of quadratic equation is a parabola. If  (‘a’ is positive) then parabola opens upwards  and its vertex is the minimum point on the graph.
If  (‘a’ is negative) then parabola opens downwards and its vertex is the maximum point on the  graph.

We recognize that  it is an upwards opening parabola.

Therefore conditions for  are;

c.
 

We have found in (a) for ;

We have found in (b) for ;

For both inequalities to be true the overlapping period of two sets of value of x is what will make  them so;

Therefore;