Compound Inequalities

And

If we have two sets A and B, then we say that an element belongs to A and B if it belongs to both of them.  For example, a student may belong to the honor society if that student is a member of A and B where

        A = the collection of all registered students

        B = the collection of all student has a GPA of at least 3.0.

Your instructor cannot belong to the honor society, since he is not a student.

 

Example

If

        A  =  {x | x > -2}

and

        B  =  {x | x <1}

then A and B is the in between part of the number line.

        A and B  =  {x | -2 < x < 1}

       

 

Exercises  

Write the following on a number line:

  1. 3x - 4 < 2 and 2x + 5 > 1

  2. 5x - 2 < 3 and 4x + 7 > 3

  3. 3x - 4 < 2 and 2x - 7 > 5


Or

If we have two sets A and B, then we say that an element belongs to A or B is it belongs to either of them.  For example, a person can get a back seat pass to a concert if that person belongs to A or B where

        A  =  set of people that know the band member personally.

        B  =  set of people that sit in the VIP seats.

We always have 

        (A and B)  C  (A or B)

 

Example:  

If

        A  =  {x | x < -1)

        B  =  {x | x > 2}

then A or B are the arrows left from -1 and right from 2.

 

 

Exercises  

Sketch the solution set on a number line.

  1. 3x + 2  >  4    or    5x + 7  <  2

  2. x - 10  >  5    or    3x + 2  <  5

  3. 4x - 2  >  2    or    5x + 1  <  11

  4. 2x - 1  <  1    or    3x  + 5  >  8

 

We also use intersection (I) for and and union (U) for or.


Double Inequalities

We can write a compound inequality as a double inequality as in the example

        -3  <  2x + 5  <  7

This can be written as a compound inequality by writing

        -3  <  2x + 5  and  2x + 5  <  7

        -8  <  2x  and  2x  <  2                    Subtracting 5

       
-4  <  x  and  x  <  1                        Dividing by 2

Exercise

Write the double inequality as a compound inequality and then solve it.

  1. -5  <  3x + 4  <  19

  2. 2y - 1  <  y + 2  <  6y + 12

  3. 15 - t  <  t + 15  <  9t - 9

  4. h + 1  <  2/3 h  <  h - 2



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