FRACTIONAL, IRRATIONAL, AND ABSOLUTE VALUE INEQUALITIES

1. FRACTIONAL INEQUALITIES


The steps to solving of  fractional inequality :


1. Find the x values  which cause the numerator and denominator are zero

2. Put them on the number line

3. Sign evey part of the number line ( either positive or negative)

4. Remember with “The denominator isn’t equal zero

5. Find the solution set


Examples :

x+2
-------   >= 0
x-1

step 1            x + 2 = 0            x – 1 = 0
                     x = -2                x  = 1
step 2                                   ●                   ○
                                            -2                  1
                                + + +             - - -              + + +

step 3                              ●                   ○
                                       -2                  1
step 4 ( x =/ 1)
step 5                               The solution set is

EXERCISES

Find the solution set of the following inequalities


2. IRRATIONAL INEQUALITIES

Irrational inequalities is inequalities with the variables are under the sign of the root.

Generally, solving of these inequalities is we must square both sides of inequalities,

We have to pay attention that the functions are under the sign of the root have to be

positive or zero (0)


Examples

1.          x – 5 < 3   (square both sides)

              x  < 14………………………………(i)

       x – 5  0   (function have to be positive or zero)

            x  ………………………………...(ii)


EXERCISES

1. Find the solution set of the following inequalities



3. ABSOLUTE VALUE INEQUALITIES

Definition : For all real numbers x  
                                                              -x , x < 0



There are 2 methods for the solving of  absolute value inequalities:

    We must square both sides
    * If              then –a < f(x) < a

    * If              then  f(x) < -a or f(x) > a

Examples

1.

    The first method:                                                    The seconds method:

    x2 – 4x + 4 > 9                                                        x -2 < -3 or  x – 2 > 3

    x2 – 4x – 5 > 0                                                            x  < -1  or     x > 5

    (x  + 1)(x-5)> 0                                           The solution set is


 + + +        - - -        + + +

              ○             ○

 -1             5

   The solution set is


2.

    The first method :                                                   The seconds method :

    (x2 – 5)2  42                                                           -4  (x2 – 5)  4

    (x2 – 5)2 – 42  0                                                     x2 – 5  4  and  x2 -5  -4

    ((x2-5) – 4)((x2-5) + 4)  0                                      x2 – 9  0  and  x2 – 1 0

    (x2 – 9)(x2 – 1)  0                                                 (x-3)(x+3)0 and (x-1)(x+1)0

    (x-3)(x+3)(x-1)(x+1)0                                        

                                                                                       + + +            - - -                  + + +

                                                                                                   ●                         ●

                                                                                                  -3                       3


 + + +    - - -           + + +     - - -       + + +                   + + +             - - -        + + +

           ●             ●            ●          ●                                                ●          ●                            

          -3            -1            1           3                                               -1          1

   The solution set is                                                  The solution set is

EXERCISES

 Find the solution set of the following inequalities :

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