1.7 - Solving Polynomial Inequalities MCB4U - Santowski

(A) Review An inequality is a mathematical statement like x+3>7, where one side of an “equation” is larger than (or smaller than or not equal to) the other side. An inequality is a mathematical statement like x+3>7, where one side of an “equation” is larger than (or smaller than or not equal to) the other side. When solving an inequality, we are solving not just for a single value for the variable, but for all possible values of the variable that satisfy the inequality. When solving an inequality, we are solving not just for a single value for the variable, but for all possible values of the variable that satisfy the inequality. These solutions for the inequality divide the domain into intervals (in this case, when x > 3, then the statement is true)

(B) Examples Solve the inequality: Solve the inequality: x > 2x x > 2x + 9. State solution in set notation and on a number line State solution in set notation and on a number line

(B) Examples Solve the polynomial inequality x > 0. Solve the polynomial inequality x > 0. Show the solution by means of a table/chart technique that takes into account the domain as it is divided into its three intervals (in this case). Present graphical solution at the same time to visually reinforce concept Show the solution by means of a table/chart technique that takes into account the domain as it is divided into its three intervals (in this case). Present graphical solution at the same time to visually reinforce concept (x + 5) (x – 5) P(x) x < -5 x < -5-ve-ve+ve -5<x<5+ve-ve-ve X > 5 +ve+ve+ve

(B) Examples Solve (x 2 -1)(x 2 +x-2)>0 algebraically. Verify graphically Solve (x 2 -1)(x 2 +x-2)>0 algebraically. Verify graphically Solve (4–x 2 )(x 2 -2x+2)<0 and verify by graphing.

(C) Examples - Applications ex 5. The population of a city is given by the model P(t) = 0.5t t where P is the population in thousands and where t = 0 represents the year When will the population be less than 330,000? ex 5. The population of a city is given by the model P(t) = 0.5t t where P is the population in thousands and where t = 0 represents the year When will the population be less than 330,000?  eqn to work with is 0.5t t < 330 which must be solved using the quadratic formula  eqn to work with is 0.5t t < 330 which must be solved using the quadratic formula

(D) Internet Links Linear Inequalities from WTAMU Linear Inequalities from WTAMU Linear Inequalities from WTAMU Linear Inequalities from WTAMU Quadratic Inequalities from WTAMU Quadratic Inequalities from WTAMU Quadratic Inequalities from WTAMU Quadratic Inequalities from WTAMU Rational Inequalities from WTAMU Rational Inequalities from WTAMU Rational Inequalities from WTAMU Rational Inequalities from WTAMU Polynomial Inequalities from PurpleMath Polynomial Inequalities from PurpleMath Polynomial Inequalities from PurpleMath Polynomial Inequalities from PurpleMath

(D) Homework Nelson text, p72, Q5-9eol,10-15,20, and for applications 19,23 Nelson text, p72, Q5-9eol,10-15,20, and for applications 19,23