SOLUTION: Solve (x + 2 < 5) ∩ (x - 7 > -6).

This problem wants you solve two inequalities separately

    x + 2 < 5      (1)
and
    x - 7 > -6     (2)

and then take the intersection of their sets of solutions.



It is the same as to solve the system of two inequalities 




OK. So, our first step is to solve inequality (1).  For it, subtract the number 2 from both sides. 
Inequality remains equivalent and takes the form

    x < 5 - 2,  
 
which is the same as

    x < 3.


So, the set of solutions to the first inequality is  { x < 3 }, or, in the interval notation  (,).


Our next step is to solve inequality (2).  For it, add the number 7 to both sides. 
Inequality remains equivalent and takes the form

    x > -6 + 7,  
 
which is the same as

    x > 1.


Thus the set of solutions to this inequality is  { x > 1 }, or, in the interval notation  (,).


The intersection of the sets  { x < 3 }  and  { x > 1} is the set  { 1 < x < 3 },  or, in the interval notation,  interval (1,3).


ANSWER.  The solution of the problem is the set  { 1 < x < 3 },  or, in the interval notation,  interval (1,3).