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# IntermediateAlgebra MATH0000D

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This 8 page Class Notes was uploaded by Aileen Davis on Tuesday October 20, 2015. The Class Notes belongs to MATH0000D at Sierra College taught by JamesDuran in Fall. Since its upload, it has received 8 views. For similar materials see /class/225378/math0000d-sierra-college in Mathematics (M) at Sierra College.

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Date Created: 10/20/15

NONLINEAR INEQUALITIES IN ONE VARIABLE A Quadratic Inequality is an inequality that can be written so that one side is a quadratic expression and the other side is 0 Examples x2 7x730 lt 0 2x2 9x7 2 2 0 First answer these questions In which quadrants are the y values positive In which quadrants are the y values negative 39T Example A Solve x2 79 gt0 This means to nd the values of x which make x279 positive greater than 0 One way to do this is to let yx2 79 graph this parabola and then nd the values of x that cause the graph to be the x axis Again x2 79 is positive when the parabola y x2 79 is the x axis The values ofx which cause the graph to be above the x axis are in interval notation We can verify this for instance let x or let x Example B Solve x2 79g 0 This means to nd the values of x which make x2 79 negative less than 0 or equal to 0 x2 79 is negative when the parabola y x2 79 is the x axis or on the x axis This occurs in quadrants and One way is to do this to let yx279 graph this parabola and then nd the values of x that cause the graph to be x axis The values ofx which cause the graph to be below or on the x axis are in interval notation We can also graph the solution We can verify this for instance let x Example C Solve x2 2x78 lt 0 This means to nd the values ofx which make x2 2x78 negative less than 0 Example D Solve x2 2x78 2 0 This means to nd the values ofx which make x2 2x78 Note from Examples C amp D the Xvalues which cause 2 to be positive are separated from the Xvalues which cause v to be negative by the Xintercepts Recall how to nd the Xintercepts set y 0 and solve for X We do not necessarily to graph the function to solve linear inequalities instead we can draw a number line representing the XaXis and keep the following in mind A region on the number line for which the value of an exgression is gositive is segarated from a region on the number line for which the value of an exgression is negative by the xintercegts Reconsider example C x2 2x 8 lt 0 When we solve x2 2x 8 0 we obtain X 4 and B These two Xvalues separate the number line the XaXis into three regions intervals 0 oo 4 4 2 2 00 Test Value TV X 5 X 0 X 3 Sign Conclusion We choose a test value TV X from each of the three regions intervals and replace the test value into x2 2x 8 If the value of x22x 8 is negative when a TV from a interval is substituted into that eXpression then x2 2x 8 will be negative for any value in that interval Therefore that region in interval notation will be part of the solution The solution in interval notation is How to Solvea uadratic Nonlinear Ine ualit 1 Replace the inequality symbol szltgt by an equal sign and set the equation equal to 0 2 Solve the equation in step 1 usually you will solve by factoring 3 Separate the number line which represents the XaXis into intervals using the solutions from step 2 For 2 Use Brackets For ltgt Use Parenthesis 4 From each interval choose a test value Ty and determine if it satis es the original ineguality You can also use the original inequality in factored form 5 If the ineguali is satis ed then the interval from Where that TV was chosen from is part of the solution If the inequality is not satis ed then the interval from Where that TV was chosen from is not part of the solution Example E Solve x2 6x 2 0 In all these problems graph your solution and put your solution in interval notation Example F Solve x3 5x2 4x 3 20 On separate paper please solve these nonlinear inequalities and graph your solutions and put your solutions in interval notation 1 Solve 2x2 lt50 2 Solve 2x2 x 6ZO 3 Solve x3x 2x 5S0 111 How to Solve a Rational Inequality 1 Replace the inequality symbol szltgt by an equal sign and set the equation equal to 0 2 Find the solutions that make both the denominator Ml numerator equal to 0 3 Separate the number line which represents the XaXis into intervals using the solutions from step 2 For 2 Use Brackets For ltgt Use Parenthesis Use parenthesis g L on the solutions that makes the denominator equal to 0 These values are not part ofthe solution 4 From each interval choose a test value Ty and determine if it satis es the original inequality or the inequality Whose right side is 0 5 If the ineguali is satis ed then the interval from Where that TV was chosen from is part of the solution If the inequality is not satis ed then the interval from Where that TV was chosen from is not part of the solution Example G Solve Pr 3 0 x x2 6 5x gt1 Example H Solve On separate paper please solve these nonlinear inequalities and graph your solutions and put your solutions in interval notation 1 4 Solve 95 430 5 Solve 3 lt2 6 Solve x5 2 2 y153 x 8y

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