Fouss math class wiki

 

Section 5 - Solving Inequalities Algebraically and Graphically

Page history last edited by Fouss 1 yr ago

 

 

 

Section 5 [Solving Inequalities] Section 5 [Solving Inequalities] Section 5 [Solving Inequalities] Section 5 [Solving Inequalities]

 

Section 5 [Solving Inequalities] Section 5 [Solving Inequalities] Section 5 [Solving Inequalities] Section 5 [Solving Inequalities]

 

Section 5 [Solving Inequalities] Section 5 [Solving Inequalities] Section 5 [Solving Inequalities] Section 5 [Solving Inequalities]

 

 

1. Solving A Basic Inequality Problem

 

     Example 1:  eq=5x-7>3x+9

 
 
       eq=2x-7>9             Combine x values  

            eq=2x>16             Combine other values

              eq=x>8                Divide both sides

 

 

                                                                  \/

 

                                        eq=<----\mid---->
 
 
                                                                 8

 

        

        *Dividing or Multiplying by a negative will flip the inequality!

 

2. Using Interval Notation: smallest ----- > biggest

 

                           eq=( 8, \infty)

 
Note: If answer is included use a bracket [
          If answer is not included use a parenthesis (
 
               
 
Example 2:  eq=-3\le6x-1<3
 
 
           eq=-2\le6x<4     Combine other values        

 

                               
   
           eq=\frac {-1}{3} \le x< \frac23            Divide
 
                   eq=[ \frac{-1}{3}, \frac 23)          Express in interval notation
 

 

 

        

 

3. Absolute Values

 

            Note: If inequality is less than ----> and

                      If inequality is great"or" than ----> or

 

             Example 3: eq=\mid x-5 \mid<2

 

 

 

                 

 

                eq=x-5<2     and     eq=x-5>-2            Remove absolute value for first ineqaulity. Flip and switch for second inequality
               
                         eq=x<7     and     eq=x>3                      Isolate variable
 
                                        eq=( 3,7)                                 Write in interval notation
 
 
                   Example 4:  eq=\mid x-5 \mid >2
               
                      Note: Use Unison ( U ) for greater than inequalities with absolute values!
 
 
                     eq=x-5>2      or    eq=x-5<-2
 
 
                              eq=x>7      or     eq=x<3
 
 
                                eq=(-\infty,3)U(7,\infty)
 
                     Note:( eq=\infty ) Infinity is never included, always use a parenthesis )

 

 
 
 
4. Critical Numbers and Powers

 

            When do we have critical #s?
            1. Inequality with powers
            2. Inequality with fraction
 
            Note: Use unison ( U ) when you have critical numbers!
 
                Example 5:   eq=2x^2+5x>12
 
               
                eq=2x^2+5x-12>0                           Get zero on one side of the inequality
 
               eq=(2x-3)(x+4)>0                              Factor
               
               

 

              eq=2x-3=0                    eq=x+4=0           Set each factor equal to zero to
 
 
               eq=x=\frac32                     eq=x=-4                            Solve for x to get critcial numbers

 

 

 
 
Critical #s:       eq=\frac32, -4
 

 

                             \/             X          \/

                        eq=<----\mid----\mid---->
                                                 -4               3/2
 
                              * Pick a number from each interval and test it in the inequality!
 
                                                
                                                eq=(-\infty,-4)U(\frac32, \infty)             Write in interval notation with unison
 

 

 

 

 

5. Critical Numbers And Fractions

 

         Steps

         1. Get zero on one side

         2. Combine Terms

         3. Critical #s: Num=0

                               Den= 0 <----- Never included!

 

 

                Example 6:   eq=\frac{2x-7}{x-5}\le3

 

 

                                     eq=\frac{2x-7}{x-5}-3\le0                Get zero on one side
 
 
                    eq=\frac{2x-7}{x-5}-3(\frac{x-5}{x-5})\le0           Multiply three by x-5 over x-5 to get a common denominator
 
 
 
                  

 

                    eq=\frac{2x-7}{x-5}-\frac{3(x-5)}{x-5}\le0                 Simplify
 
 
 
                     eq=\frac{2x-7}{x-5}-\frac{3x-15}{x-5}\le0                    Distribute
 
 
 
 
 
                  eq=\frac{2x-7}{x-5}+\frac{-3x+15}{x-5}\le0                  Distribute the negative to the numerator of the second fraction
 
 
 
 
 
                       eq=\frac{2x-7-3x+15}{x-5}\le0                       Add 
 
 
 
                                              eq=\frac{-x+8}{x-5}\le0                      Combine like terms
 
 
 
                     eq=-x+8=0                  eq=x-5=0             Set numerator and denominator equal to zero
 
 
 
                                  eq=x=8                            eq=x=5                 Solve for x to get critical numbers
 
 
 
Critical #s: eq=5,8
 
 
                            \/                  X           \/
                     eq=<----\mid----\mid---->
                                               5                     8
 
 
 
                                             eq=(-\infty,5)U[8,\infty)                  Write in interval notation with unison
 
 
 
 
 
 
 
 
                         
 
 
 

 

 

 

 

 

 

 

 

 

Comments (1)

Fouss said

at 8:16 pm on Jan 8, 2008

Nice job with the graphs... but can you change "unison" to union? 20/20

You don't have permission to comment on this page.

Printable version
 
Anonymous Create an account or Log in