How can a compound inequality have a solution of all real numbers but only one exception ?

Question

Explain why the solution of 5x – 3 > 14.5 or < 4 has a solution of all real numbers, with one exception.

Answer 1

"The solution" to these two inequalities is the value(s) of x which make the combination of the inequalities true.  In this problem, the conjunction "OR" is used.  That means a value of x that makes either inequality (or both) true is a solution.

 

When is 5x-3>14.5 true?

     5x > 17.5       [add +3 to both sides and retain sense of inequality]

      x > 3.5         [divide both sides by +5 and retain sense of inequality]

 

When is x<4 true?

      x<4                     [duh]

 

When is (x>3.5) OR (x<4) true?

     for all real numbers x

 

 

Now, the compound inequality (5x -3 > 17) OR (x < 4) [check it out] is true for all real numbers x with the exception of x=4 (because neither expression includes "=").