ࡱ> \^ [g 5bjbjVV .<<g-m&&iiiii}}}8l!T}8u9%9%9%9%,,,B8D8D8D8D8D8D899<D8i,F+,,,D8ii9%9% Y8777,Fi9%i9%B87,B87779%`tl$}b,N7.8o8087<-N <7<i70,,7,,,,,D8D87,,,8,,,,<,,,,,,,,,& /: LESSON 1 REVIEW OF SOLVING NONLINEAR INEQUALITIES In this lesson, we will make use of the Axiom of Trichotomy given below. Axiom of Trichotomy A real number can only be one of the following: positive, negative, or zero. NOTE: When you substitute a real number in for the variable in a nonlinear expression, you will either get another real number (which is either positive, negative, or zero) or something that is undefined as a real number. For example, when we replace the x in the nonlinear expression  EMBED Equation.3  by 4, we get the real number  EMBED Equation.3  obtained by  EMBED Equation.3 . The resulting real number is negative. If we replace the x by  EMBED Equation.3  in the expression, we get the real number  EMBED Equation.3  obtained by  EMBED Equation.3 . This number is positive. If you replace the x by  EMBED Equation.3  in the expression, you will get the real number zero obtained by  EMBED Equation.3 . Finally, if we replace the x by 3, we get an undefined real number since we get division by zero obtained by  EMBED Equation.3 . Thus, to solve a nonlinear inequality, we will find all the real numbers that make a nonlinear expression equal to zero. We will also have to find all the numbers that make the nonlinear expression undefined. Thus, all the remaining real numbers, when substituted for the variable in the nonlinear expression, would make the resulting real number either be positive or negative. Thus, a nonlinear expression has the ability to change signs at the real numbers where the expression is either zero or undefined. We will determine when a nonlinear expression is positive and negative using the following three steps: Step 1 Find all the real numbers that make the nonlinear expression equal zero and all the real numbers that make the expression undefined. Step 2 Plot all the numbers found in Step 1 on the real number line. Step 3 Using the real number line in Step 2, identify the open intervals determined by the plotted numbers. For each open interval, pick a real number that is in the interval. We will call this number the test value for the interval. Substitute the test value for the variable in the nonlinear expression. Whatever sign the expression has for this test value, the expression will have the same sign for any number in the open interval. Examples Solve the following nonlinear inequalities. 1.  EMBED Equation.3  Step 1: Find when the nonlinear expression  EMBED Equation.3  is equal to zero. That is, solve the equation  EMBED Equation.3 .  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Find when the nonlinear expression  EMBED Equation.3  is undefined. The expression  EMBED Equation.3  is defined for all real numbers x. Step 2: Plot all the numbers found in Step 1 on the real number line. +  EMBED Equation.3  + Sign of  EMBED Equation.3   ( (  EMBED Equation.3  8 Step 3: Use the real number line to identify the open intervals determined by the plotted numbers. Pick a test value for each open interval. Interval Test Value Sign of  EMBED Equation.3  =  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  0  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  9  EMBED Equation.3   EMBED Equation.3  Answer:  EMBED Equation.3  2.  EMBED Equation.3  NOTE: This is a two part problem. One part of the problem is to solve the nonlinear inequality  EMBED Equation.3 . The other part of the problem is to solve the equation  EMBED Equation.3 . We will use the three step method to solve the nonlinear inequality  EMBED Equation.3 : Step 1: Find when the nonlinear expression  EMBED Equation.3  is equal to zero. That is, solve the equation  EMBED Equation.3 . The fraction is equal to zero if and only if the numerator of the fraction is equal to zero. That is,  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Find when the nonlinear expression  EMBED Equation.3  is undefined. The fraction is undefined if and only if the denominator of the fraction is equal to zero. That is,  EMBED Equation.3  undefined EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Step 2: Plot all the numbers found in Step 1 on the real number line.  EMBED Equation.3  +  EMBED Equation.3  Sign of  EMBED Equation.3   ( (  EMBED Equation.3  6 Step 3: Use the real number line to identify the open intervals determined by the plotted numbers. Pick a test value for each open interval. Interval Test Value Sign of  EMBED Equation.3   EMBED Equation.3  0  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  5  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  7  EMBED Equation.3   EMBED Equation.3  Thus, the solution for the nonlinear inequality  EMBED Equation.3  is the set of real numbers given by  EMBED Equation.3 . The solution for  EMBED Equation.3  was found in Step 1 above. Thus, the solution for  EMBED Equation.3  is the set  EMBED Equation.3 . Putting these two solutions together, we have that the solution for  EMBED Equation.3  is the set of real numbers  EMBED Equation.3 . Answer:  EMBED Equation.3  3.  EMBED Equation.3  Step 1:  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  The expression  EMBED Equation.3  is defined for all real numbers t. Step 2: +  EMBED Equation.3  + Sign of  EMBED Equation.3   ( (  EMBED Equation.3   EMBED Equation.3  Step 3: Interval Test Value Sign of  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  0  EMBED Equation.3   EMBED Equation.3  Answer:  EMBED Equation.3  4.  EMBED Equation.3  Step 1:  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  undefined EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Step 2: +  EMBED Equation.3  +  EMBED Equation.3  Sign of  EMBED Equation.3   ( ( (  EMBED Equation.3   EMBED Equation.3  3 Step 3: Interval Test Value Sign of  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  0  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  2.7  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  4  EMBED Equation.3   EMBED Equation.3  Answer:  EMBED Equation.3  5.  EMBED Equation.3  NOTE: This is a two part problem. One part of the problem is to solve the nonlinear inequality  EMBED Equation.3 . The other part of the problem is to solve the equation  EMBED Equation.3 . We will use the three step method to solve the nonlinear inequality  EMBED Equation.3 : Step 1:  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  The expression  EMBED Equation.3  is defined for all real numbers y. Step 2:  EMBED Equation.3  +  EMBED Equation.3  Sign of  EMBED Equation.3   ( (  EMBED Equation.3   EMBED Equation.3  Step 3: Interval Test Value Sign of  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  0  EMBED Equation.3   EMBED Equation.3  2  EMBED Equation.3  Thus, the solution for the nonlinear inequality  EMBED Equation.3  is the set of real numbers given by  EMBED Equation.3 . The solution for  EMBED Equation.3  was found in Step 1 above. Thus, the solution for  EMBED Equation.3  is the set  EMBED Equation.3 . Putting these two solutions together, we have that the solution for  EMBED Equation.3  is the set of real numbers  EMBED Equation.3 . Answer:  EMBED Equation.3  6.  EMBED Equation.3  NOTE: Since  EMBED Equation.3 , then  EMBED Equation.3  Step 1:  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  undefined EMBED Equation.3   EMBED Equation.3  Step 2: +  EMBED Equation.3   EMBED Equation.3  + Sign of  EMBED Equation.3   ( ( (  EMBED Equation.3  0  EMBED Equation.3  Step 3: Interval Test Value Sign of  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  1  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  3  EMBED Equation.3   EMBED Equation.3  Answer:  EMBED Equation.3  7.  EMBED Equation.3  NOTE: This is a two part problem. One part of the problem is to solve the nonlinear inequality  EMBED Equation.3 . The other part of the problem is to solve the equation  EMBED Equation.3 . We will use the three step method to solve the nonlinear inequality  EMBED Equation.3 : Step 1:  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  undefined EMBED Equation.3   EMBED Equation.3  Step 2: Sign of  EMBED Equation.3 :  EMBED Equation.3  + +  EMBED Equation.3  + +  ( ( ( ( (  EMBED Equation.3   EMBED Equation.3  0  EMBED Equation.3  7 Step 3: Interval Test Value Sign of  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  0.1  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  1  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  8  EMBED Equation.3   EMBED Equation.3  Thus, the solution for the nonlinear inequality  EMBED Equation.3  is the set of real numbers  EMBED Equation.3 . The solution for  EMBED Equation.3  was found in Step 1 above. Thus, the solution for  EMBED Equation.3  is the set  EMBED Equation.3 . Putting these two solutions together, we have that the solution for  EMBED Equation.3  is the set of real numbers  EMBED Equation.3 . Answer:  EMBED Equation.3      Copyrighted by James D. 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