Ch 3.5 – Polynomial and Rational Inequalities                                                                                                                 DAY 6 
                                                                               
            HW:  Page 193  #7, 9, 11, 15, 17, 19, 25, 29, 31, 33, 35, 37, 45, 57, 59 
             
            Pre-Calculus Warm-Up (before 3.5) 
             
            Solve for x: 
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            Add or subtract the following fractions: (HINT: you need a common denominator!) 
             
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            Graph the following: 
                  5.      y =(x !3)2(x +2)                                                                  6.  y = 2x !4                                    
                                                                                                                         x +3
            (HINT: remember end behavior & multiplicity of zeros!) 
             
             
             
                                          y                                                                                          y 
             
             
             
             
             
                                                                       x                                                                                         x 
             
             
             
             
             
             
             
             
            Intervals above the x-axis: ___________________                                                 Intervals above the x-axis: ___________________ 
             
            Intervals below the x-axis: ___________________                                                 Intervals below the x-axis: ___________________ 
                                                        
               
               
              Remember Interval Notation –  
                                             can be written                     can be graphed                      (                      )    
                                                                                                                             a                     b 
               
                                             can be written                      can be graphed                         [                         ] 
                                                                                                                              a                   b 
               
                                     or              can be written                                         can be graphed                             )           [ 
                                                                                                                                                              a        b 
                     
              Review of Algebra I:  Solve the inequality and graph the solution set.  Write answers in both set and interval notation. 
               
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              To solve a polynomial inequality with degree higher than one: 
                    •      Rearrange the inequality so that it is of the form:                                         ,                  ,                 , or                     
                           (Polynomial on one side of the inequality; with zero on the other side) 
                    •      Find the “critical numbers” by finding where f (x) = 0  (basically, find the zeros of the polynomial…you’ve done this!) 
                    •      Use the critical numbers you found to divide the real number line into regions 
                    •      Test each region: evaluate the original inequality using a number within the region 
                    •      Write the solution set using interval notation (be careful when choosing hard or soft brackets!) 
               
              Let’s solve a polynomial inequality, using the polynomial from the warm up, but this time without graphing. 
              (x !3)2(x +2)>0 
               
              Critical Numbers:  ______________________                                                                Number line: ___________________________________ 
               
               
               
               
              Solution set: ________________________ 
              (Does this match the answer we got in the warm up?) 
               
              Examples.  Solve the inequality and graph the solution set. 
              3.                                                                                                       4.                               
               
               
               
                                                          
            
           To solve a rational inequality: 
                 •    Rearrange the inequality so that it is of the form:                          ,              ,               , or               
                      (Ratio on one side of the inequality; with zero on the other side) 
                 •    Rewrite the expression as ONE fraction 
                 •    Find the “critical numbers” – TWO PARTS THIS TIME: Find where f (x) = 0 and where f (x) is undefined 
                      (zeros…and vertical asymptote! We’ve done this before, too!) 
                 •    Use the critical numbers to divide the real number line into regions 
                 •    Test each region: evaluate the original inequality using a number within the region 
                 •    Write the solution set using interval notation (be careful when choosing hard or soft brackets) 
            
            
           Let’s solve a rational inequality, using the rational function from the warm up, but this time without graphing. 
            
            2x !4
                      "0 
             x +3
            
            
            
            
            
            
           Critical Numbers:  ______________________                                    Number line: _____________________________________________ 
            
            
            
            
           Solution set: ________________________ 
           (Does this match the answer we got in the warm up?) 
            
                                                       
            Solve the inequality.  
             
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