Solve Rational Inequalities - Tutorial

Example 1 Solve the rational inequality given by

Solution to Example 1

• The zero of x - 1 is 1.
  
  
• The zero of x + 2 is -2.
  
  
• The two zeros divide the real number line into three intervals (Note that the zeros are ordered from smallets to largetst).
  
  
  (- infinity , -2)
  (-2 , 1)
  (1 , + infinity)
  
  
  
• You now chose test values within each interval and test the rational expression ( x - 1) / (x + 2) to find its sign.
  
  
  a) (- infinity , -2)
  
  • test value x = -3
  • We now evaluate ( x - 1) / (x + 2) at x = -3 to find its sign.
  • ( x - 1) / (x + 2)
    = (-3 - 1) / (-3 + 2)
    = 4 which is positive.
  
  
  
  b) b) (-2 , 1)
  
  • test value x = 0
  • We now evaluate ( x - 1) / (x + 2) at x = 0 to find its sign.
  • ( x - 1) / (x + 2)
    = (0 - 1) / (0 + 2)
    = -1/2 which is negative.
  
  
  
  c) (1 , + infinity)
  
  • test value x = 2
  • We now evaluate ( x - 1) / (x + 2) at x = 2 to find its sign.
  • ( x - 1) / (x + 2)
    = (2 - 1) / (2 + 2)
    = 1/4 which is positive.
  Let us now put all the above results in a table
  
  
  • In the interval (- infinity , -2) , ( x - 1) / (x + 2) is positive
    
  • In the interval (- 2 , 1) , ( x - 1) / (x + 2) is negative
    
  • In the interval (1 , + infinity) , ( x - 1) / (x + 2) is positive
    
  
  
  

  	-2		1
  +	undefined	-	0	+

  
  
  
• Conclusion The solution set of the given rational inequality is given by the interval
  
  (- infinity , -2) U (1 , + infinity)

Matched Exercise Solve the rational inequality given by

Example 2  Solve the rational inequality given by

Solution to Example 2

• NOTE: Do not multiply both sides by the LCD (x - 3)(x + 4) as you would do if this was an equation. The sign of (x - 3)(x + 4) changes with x and we do not know if the order of the inequality is to be changed or not.
• Given
  
  2 / (x - 3) < 3 / (x + 4)
  
  
  
• Write the inequality with the right side equal to zero by subtracting
  3 / (x + 4) from both sides.
  
  2 / (x - 3) - 3 / (x + 4) < 0
  
  
  
• Rewrite the inequality so that the two terms making the left side have the same denominator.
  
  2(x + 4)/ [ (x - 3)(x + 4) ] - 3(x - 3) / [ (x + 4)(x - 3) ] < 0
  
  
  
• Multiply factors and add the two rational expressions on the left side of the inequality
  
  ( 2x + 8 - 3x + 9) / [ (x - 3)(x + 4) ] < 0
  
  
  
• Group like terms in the numerator
  
  (-x + 17) / [ (x - 3)(x + 4) ] < 0
  
  
  
• The zero of - x + 17 is 17
  
• The zeros of (x - 3)(x + 4) are 3 and - 4
  
  
• The three zeros divide the real number line into four intervals (Note that the zeros are ordered from the smallest to the largest).
  
  (-infinity , -4) , (-4 , 3) , ( 3 , 17) , (17 , +infinity)
  
  
  
  • a) (- infinity , -4) test value: x = -5
    We now evaluate ( -x + 17) / [ (x - 3)(x + 4) ] at x = -5 to find its sign.
    ( -x + 17) / [ (x - 3)(x + 4) ]
    = ( -(-5) + 17) / [ (-5 - 3)(-5 + 4) ]
    = 22/8
  
  
  • b) (- 4 , 3) test value: x = 0
    We now evaluate ( -x + 17) / [ (x - 3)(x + 4) ] at x = 0 to find its sign.
    ( -x + 17) / [ (x - 3)(x + 4) ]
    = ( 0 + 17) / [ (0 - 3)(0 + 4) ]
    = -17 / 12
  
  
  • c) (3 , 17) test value: x = 4
    We now evaluate ( -x + 17) / [ (x - 3)(x + 4) ] at x = 4 to find its sign.
    ( -x + 17) / [ (x - 3)(x + 4) ]
    = (- 4 + 17) / [ (4 - 3)(4 + 4) ]
    = 13/8
  
  
  • d) (17 , +infinity) test value: x = 18
    We now evaluate ( -x + 17) / [ (x - 3)(x + 4) ] at x = 18 to find its sign.
    ( -x + 17) / [ (18 - 3)(x + 4) ]
    = ( -18 + 17) / [ (18 - 3)(18 + 4) ]
    = -1/330
  
  
  
• Let us now put all the above results in a table
  
  

  	-4		3		17
  +	undefined	-	undefined	+	0	-

  
  
  

  Conclusion The solution set of the given rational inequality is given by the interval

  (- 4 , 3) U (17 , + infinity)

Matched Exercise Solve the rational inequality given by

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