Download Solving Polynomial and Rational Inequalities: A Step-by-Step Guide - Prof. Stephanie L. Ho and more Exams Algebra in PDF only on Docsity! 1 MATH 1111-College Algebra Polynomial and Rational Inequalities Polynomial Inequalities b Any polynomial equation can be written as a polynomial inequality by replacing the = symbol with an inequality symbol (โค,โฅ,<,>) b For example the quadratic equation 3x2-2x-5=0 can be written as the following quadratic inequality: 3x2-2x-5>0 b The solutions (roots) to the quadratic equation would be the zeros of the function b The solutions to the quadratic inequality would be all x values for which the curve is above the x-axis (for > 0 in the above example) For Evaluation Only. Copyright (c) by Foxit Software Company, 2004 Edited by Foxit PDF Editor 2 Polynomial Inequality continued b Move terms until 0 is on one side of the inequality b Factor the inequality b Find the critical values (x-intercepts) b Construct a sign table b Choose the intervals where the inequality meets the condition Solve 3x roots of are Now construct sign table using intervals Where the function value is > 0 Choose that interval for the solution to the inequality 3x 2 2 โ โ > โ + > โ + = = = โ โโ โ โ โ โ โ > 2 5 0 3 5 1 0 3 5 1 0 5 3 1 1 1 5 3 5 3 2 5 0 x x x x x x x x ( )( ) ( )( ) , ( , )( , )( , ) Construct Sign Table Int. Test Value 3x-5 X + 1 F (x) ( , )โโ โ1 ( , )โ1 5 3 ( , )53 โ 5 Constructing Sign Table Int. Test Value X+1 X -2 F (x) ( , )โโ โ1 -2 -1 -4 1/4 ( , )โ12 0 1 -2 -1/2 ( , )2 โ 3 4 1 4 Solution to Rational Inequality b Choose the intervals where the function value <0 and b also choose the value where function = 0 which is the x-intercept -1 b Solution is Int. Test Value X+1 X -2 F (x) ( , )โโ โ1 -2 -1 -4 1/4 ( , )โ12 0 1 -2 -1/2 ( , )2 โ 3 4 1 4 x x + โ โค 1 2 0 [ , )โ1 2 6 Rational Inequality-Example 2 b Solve b First get 0 on one side and make fractions into one fraction b Find critical values b Make sign table b Write solution to inequality 4 4 0 4 4 0 4 4 1 0 4 4 0 2 2 2 0 0 2 โฅ + โฅ + โ โฅ + โ โข โฅ โ + โฅ โ โ โ = = x x x x x x x x x x x x x x x x ( ) ( )( ) x - intercept 2, vert. asmyp. critical values 0 and 2 4 4 โฅ + x x Constructing sign table Int. Test Val. X-2 X -2 X F (x) ( , )โโ 0 ( , )02 ( , )2 โ 7 Constructing sign table Int. Test Val. X-2 X -2 X F (x) ( , )โโ 0 -1 -3 -3 -1 -9 ( , )02 1 -1 -1 1 1 ( , )2 โ 3 1 1 3 1/3 Solution to Rational Inequality b Choose the intervals where f(x) <0 and also where f(x)=0, which is at x-intercept 2 b Write solution as a union of the two intervals: Int. Test Val. X-2 X -2 X F (x) ( , )โโ 0 -1 -3 -3 -1 -9 ( , )02 1 -1 -1 1 1 ( , )2 โ 3 1 1 3 1/3 ( , ) [ , ]โโ 0 2 2U
