2Solving Inequalities Algebraically and Graphically Polynomial InequalitiesWhen solving a polynomial inequality, just like the polynomial equation, move everything to one side so it is greater than or less than 0 . This tells us whether the solutions being looked for are values that give negative or positive results.Next, we will need to locate what is known as critical numbers. These are values that create zeros for the polynomial inequality, and when placed in order, divide the number line into “test” sections.This will create test intervals for the inequality which in turnleads to the solution for the polynomial inequality.A better understanding will follow by looking at an example.
3Solving Inequalities Algebraically and Graphically Polynomial InequalitiesExample: SolveStep 1: Place the inequalityin standard form and factor.Step 2: Find the criticalnumbers by setting thefactors equal to zero andsolving.CRITICAL NUMBERS
4Solving Inequalities Algebraically and Graphically Polynomial Inequalities (cont.)Why were opencircles used?Step 3: Place the critical numbers on the number line.Notice how the number line is now divided into three sections.Section 1Section 2Section 3
5Solving Inequalities Algebraically and Graphically Polynomial Inequalities (cont.)Step 4: Test each section back into the factored form of the polynomialinequality and place the “sign” of the product over that section.++Test – 5Test 0Test 2( – )( – ) ( + )( – )( + ) ( – )( + )( + ) ( + )Step 5: State the solution using ( ) or [ ] or combination of both.Because these were the intervals wherepositive results were located on the number line.
6Solving Inequalities Algebraically and Graphically Try this.Determine the solution forStep 1:Step 3: Graph the critical numbersStep 4: Test sections–+–+Step 2: Critical NumbersStep 5: Write the solution.
7Solving Inequalities Algebraically and Graphically Solving Rational InequalitiesSimplify the rational inequality into one expression. Critical numbers will include the zeros and the undefined values.Ex.: Determine the solution forStep 1:Step 2: Critical Numbers – set thenumerator and denominator equalto zero and solve.* Undefined value
8Solving Inequalities Algebraically and Graphically Solving Rational Inequalities (cont.)Why these symbols?Step 3: Graph the critical numbers.–+–Step 4: Test each section using the simplified standard form.Test 4Test 6Test 9Step 5: State the solution.Why is a combination ( ] used?
9Solving Inequalities Algebraically and Graphically Try this.Solve the inequalitySolution:Critical NumbersTesting:Test -2Test 0Test 5–+–(–1, 4)Solution:
10Solving Inequalities Algebraically and Graphically This technique can also be applied to finding the domain of a function.Example: Find the domain forStep 1: Write an inequality that reflects the problem and then simplify.Why is this inequality used?Step 3: Graph and determinethe signs of the intervals.–+–Step 4: State the domain basedon the interval(s).Step 2: Find the critical numbers[–8, 8]