Inequality

Motivation:

                       The relationship between two quantities is with other possibilities other than “equal”.  For example,   sometimes we hear “8 is larger than 2” or “1 is less than 3”. Similarly, we would like to have mathematical language to describe it:

 

                         8 > 2      ;  8 is larger than 2 .

                         1 < 3     ;    1 is less than 3.

                        1  b     ;    1 is less than or equal to b .

                        b  2     ;     b is larger than or equal to 2 .

 

                   Then, we check if we can observe some properties similar to equalities:

 

                                  8 > 2

                                 8+2 > 2+2            ; add both sides by 2

                                  8  5 > 2  5      ; multiply both sides by 5

                         

                       How about multiply both sides by (-1) ?

                      

                                            8  (-1) = -8 ,   2  (-1) = -2

                       And

-8 < -2

                        Originally,  we have     8 > 2 .

                        The relationship between left side and right side is “reversed”.

 

                        How about absolute value  associated with inequality ?  Let’s check:

 

                              If  |x| < 3, then   -3< x < 3 .

 

                        What if  |x| > 3 ?  Where is the range for x that satisfies the inequality?

                         The answer is

                                                    x > 3  or x < -3 .

 

                        We summarize as follows:

 

Property :   The direction of “inequality” is reversed if both sides are multiplied by a negative number.   In other words,

 

                             If   b > a   and c < 0 ,

                             then         cb < ca .

 

Property:    if  |x| < a  ( where a > 0 ),  then

                                 -a < x < a  .

 

Property:    if  |x| > a  ( where a > 0 ),  then

                               x > a or x < -a .

 

Example : Solve  x + 2 > 5 .

                  The solution is    x > 3 .

 

 

Example:    | x-2| < 5 .  Find the range of x that satisfies the inequality.

             Sol

                           -5 < x-2 < 5

                          -5+2 < x < 5+2

                    Then      -3 < x < 7  .