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Appendix B.4 Solving Inequalities Algebraically And Graphically

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**Solving Inequalities Algebraically and Graphically**

Polynomial Inequalities When 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 turn leads to the solution for the polynomial inequality. A better understanding will follow by looking at an example.

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**Solving Inequalities Algebraically and Graphically**

Polynomial Inequalities Example: Solve Step 1: Place the inequality in standard form and factor. Step 2: Find the critical numbers by setting the factors equal to zero and solving. CRITICAL NUMBERS

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**Solving Inequalities Algebraically and Graphically**

Polynomial Inequalities (cont.) Why were open circles used? Step 3: Place the critical numbers on the number line. Notice how the number line is now divided into three sections. Section 1 Section 2 Section 3

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**Solving Inequalities Algebraically and Graphically**

Polynomial Inequalities (cont.) Step 4: Test each section back into the factored form of the polynomial inequality and place the “sign” of the product over that section. + + Test – 5 Test 0 Test 2 ( – )( – ) ( + ) ( – )( + ) ( – ) ( + )( + ) ( + ) Step 5: State the solution using ( ) or [ ] or combination of both. Because these were the intervals where positive results were located on the number line.

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**Solving Inequalities Algebraically and Graphically**

Try this. Determine the solution for Step 1: Step 3: Graph the critical numbers Step 4: Test sections – + – + Step 2: Critical Numbers Step 5: Write the solution.

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**Solving Inequalities Algebraically and Graphically**

Solving Rational Inequalities Simplify the rational inequality into one expression. Critical numbers will include the zeros and the undefined values. Ex.: Determine the solution for Step 1: Step 2: Critical Numbers – set the numerator and denominator equal to zero and solve. * Undefined value

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**Solving 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 4 Test 6 Test 9 Step 5: State the solution. Why is a combination ( ] used?

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**Solving Inequalities Algebraically and Graphically**

Try this. Solve the inequality Solution: Critical Numbers Testing: Test -2 Test 0 Test 5 – + – (–1, 4) Solution:

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**Solving Inequalities Algebraically and Graphically**

This technique can also be applied to finding the domain of a function. Example: Find the domain for Step 1: Write an inequality that reflects the problem and then simplify. Why is this inequality used? Step 3: Graph and determine the signs of the intervals. – + – Step 4: State the domain based on the interval(s). Step 2: Find the critical numbers [–8, 8]

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