Wavy Line Method Application - Complex Algebraic Inequalities

WAVY LINE METHOD APPLICATION COMPLEX ALGEBRAIC INEQUALITIES

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Table of Contents Introduction ............................................................................................................................................................................... 2 Wavy Line Method Application - Multiple Instances of the Same Root ............................................................ 3 How to draw the wavy line? ........................................................................................................................................... 4 Using Wavy Line to Solve the Inequality................................................................................................................... 5 Food for Thought ................................................................................................................................................................ 6 What if the Algebraic Expression contains a Fraction?....................................................................................... 6 Takeaway .................................................................................................................................................................................... 7

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INTRODUCTION Many of you are familiar with the Wavy Line Method used to solve inequalities containing algebraic expressions in single variable. This method helps identify the range of values satisfying an inequality. For instance, consider the following question: Find the range of values of x satisfying the inequality (x – 3) (x – 6) < 0.

To solve this inequality, you draw the wavy line as follows:

And your inferences regarding the expression (x – 3)(x – 6) would be: 1. The expression will be positive for the range of values for which the curve is above the number line. a. In the above example, the pertinent ranges are x < 3 and x > 6. So, any value of x either less than 3 or greater than 6 will make the above expression positive in sign. 2. The expression will be equal to zero for the values at which the curve intersects the number line. a. In the above example, the pertinent points are x = 3 and x = 6. So, both these values make the expression zero. 3. The expression will be negative for the range of values for which the curve is below the number line. a. In the above example, there is only one portion where the curve is below the number line and this portion corresponds to the range 3 < x < 6. So, any value of x strictly between 3 and 6 will make the above expression negative in sign. Since the above question asks for the range of values of x for which the expression is negative, the answer would be 3 < x < 6.

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Well, that was a simple question, so application of the wavy line method was fairly straightforward. Now, for many test takers, things can seem complicated as exponents and algebraic fractions are added into the mix. However, the application of this method is pretty straightforward in such scenarios too. In this article we’ll be focusing on such complex application of the wavy line method and list out the two fundamental rules that are used to draw the wavy line to solve any inequalities testing such algebraic expressions on the GMAT. (This article benefits those who are not familiar with the wavy line method. The rules provided in this article are generic and can be utilized for all cases. )

WAVY LINE METHOD APPLICATION - MULTIPLE INSTANCES OF THE SAME ROOT Try to solve the following inequality using the Wavy Line Method: (x−1)2(x−2)(x−3)(x−4)3