1.7 - Solving 1.7 - Solving Polynomial InequalitiesPolynomial Inequalities

MCB4U - SantowskiMCB4U - Santowski

(A) Review(A) Review An inequality is a An inequality is a

mathematical statement mathematical statement like x+3>7, where one like x+3>7, where one side of an “equation” is side of an “equation” is larger than (or smaller larger than (or smaller than or not equal to) the than or not equal to) the other side.other side.

When solving an When solving an inequality, we are solving inequality, we are solving not just for a single value not just for a single value for the variable, but for all for the variable, but for all possible values of the possible values of the variable that satisfy the variable that satisfy the inequality. inequality.

These solutions for the These solutions for the inequality divide the inequality divide the domain into domain into intervals intervals (in (in this case, when x > 3, this case, when x > 3, then the statement is then the statement is true)true)

(B) Examples(B) Examples

Solve the inequality:Solve the inequality: -3 - 4x -3 - 4x >> 2x + 9. 2x + 9.

State solution in set State solution in set notation and on a notation and on a number linenumber line

  

(B) Examples(B) Examples Solve the polynomial Solve the polynomial

inequality xinequality x22 - 25 > 0. - 25 > 0.

Show the solution by Show the solution by means of a table/chart means of a table/chart technique that takes into technique that takes into account the domain as it account the domain as it is divided into its three is divided into its three intervals (in this case). intervals (in this case). Present graphical solution Present graphical solution at the same time to at the same time to visually reinforce conceptvisually reinforce concept

(x + 5)(x + 5) (x – 5)(x – 5) P(x)P(x)

x < -5x < -5 -ve-ve -ve-ve +ve+ve

--5<x<5<x<55

+ve+ve -ve-ve -ve-ve

X > 5X > 5 +ve+ve +ve+ve +ve+ve

(B) Examples(B) Examples

Solve (xSolve (x22-1)(x-1)(x22+x-+x-2)2)>>0 algebraically. 0 algebraically. Verify graphicallyVerify graphically

SolveSolve

(4–x(4–x22)(x)(x22-2x+2)<0 -2x+2)<0

and verify by graphing.and verify by graphing.

(C) Examples - (C) Examples - ApplicationsApplications

ex 5. The population of a city is given ex 5. The population of a city is given by the model P(t) = 0.5tby the model P(t) = 0.5t22 + 10t + 200 + 10t + 200 where P is the population in thousands where P is the population in thousands and where t = 0 represents the year and where t = 0 represents the year 2000. When will the population be less 2000. When will the population be less than 330,000?than 330,000?

eqn to work with is 0.5teqn to work with is 0.5t22 + 10t + 200 < + 10t + 200 < 330 which must be solved using the 330 which must be solved using the quadratic formulaquadratic formula

(D) Internet Links(D) Internet Links

Linear Inequalities from WTAMULinear Inequalities from WTAMU Quadratic Inequalities from Quadratic Inequalities from

WTAMUWTAMU Rational Inequalities from WTAMURational Inequalities from WTAMU Polynomial Inequalities from Polynomial Inequalities from

PurpleMathPurpleMath

(D) Homework(D) Homework

Nelson text, p72, Q5-9eol,10-Nelson text, p72, Q5-9eol,10-15,20, and for applications 19,2315,20, and for applications 19,23