## Solving Algebraically- Polynomial Inequalities - Section 3: Solving Polynomial Inequalities Algebraically

*Solving Algebraically- Polynomial Inequalities*

# Polynomial Inequalities

Lesson 12 of 15

## Objective: Students will be able to find the solution to a polynomial inequality.

*50 minutes*

#### Warm up and Homework Review

*10 min*

I include **Warm ups** with a **Rubric** as part of my daily routine. My goal is to allow students to work on **Math Practice 3** each day. Grouping students into homogeneous pairs provides an opportunity for appropriately differentiated math conversations. To prepare students for the lesson. this warm up has students refresh what they know about linear inequalities by as they determine whether or not the problem presented has been solved correctly. Please watch my Video Narrative for more information.

I also use this time to correct and record any past Homework.

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This introductory lesson is an extension that goes beyond the topics required in Algebra 2 by the Common Core. It is in no way exhaustive of this topic but instead gives students a conceptual starting place.

I chose to start this lesson graphically to lay the conceptual groundwork for polynomial inequalities. My students are already familiar with solving linear inequalities graphically so this idea isn't new. This will help them see why the solution may be more than one interval.

This lesson begins with the inequality: x^{2} – 1 > x + 5. The first thing the students do is graph each side of this inequality using their graphing calculator. I ask them to locate the solution and we verbalize the solution. I then have them write the solution using interval notation and graph it on a number line. All inequalities we solve will be represented both of these ways. Next, I have them rewrite this inequality into equivalent forms, like x^{2} > x + 6, which we make a list of on the board, and they choose one or two to graph on their calculator. They will quickly see that they get the same solution no matter which form we use (**Math Practice 7**). We will then do a think-pair-share discussing which form made it the easiest to find the solution (**Math Practice 5**). Next, I will give them the same inequality with a switched sign.

There are several more polynomial inequalities for the students to solve graphically. Each one provides students with a slightly different variation. Specifics about these inequalities are located in the PowerPoint.

#### Resources

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Now that the students have a good idea of what the solutions to these types of equations look like, we will solve them Algebraically. I begin with the original problem x^{2} – 1 > x + 5 and model its solution. We then do **Guided Practice** on several more problems including the ones we solved graphically.

As we go through this, I ask the students to think about why we would use this method rather than the graphing method.

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#### Exit Ticket

*2 min*

I use an exit ticket each day as a quick formative assessment to judge the success of the lesson.

Today's Exit Ticket asks students to find the solution to a polynomial inequality.

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This Assignment has two polynomial inequalities to be solved graphically and six to be solved graphically. The goal of this section is to reinforce the skills learned in the day's lesson. It ends by asking the students to identify with method they prefer and why (**Math Practice 5**).

This assignment was created with Kuta Software, an amazing resource for mathematics teachers.

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- UNIT 1: Modeling with Expressions and Equations
- UNIT 2: Modeling with Functions
- UNIT 3: Polynomials
- UNIT 4: Complex Numbers and Quadratic Equations
- UNIT 5: Radical Functions and Equations
- UNIT 6: Polynomial Functions
- UNIT 7: Rational Functions
- UNIT 8: Exponential and Logarithmic Functions
- UNIT 9: Trigonometric Functions
- UNIT 10: Modeling Data with Statistics and Probability
- UNIT 11: Semester 1 Review
- UNIT 12: Semester 2 Review

- LESSON 1: Laws of Exponents
- LESSON 2: Sorting Polynomial Equations and Identities
- LESSON 3: Operations with Polynomials Day 1 of 2
- LESSON 4: Operations with Polynomials Day 2 of 2
- LESSON 5: Products of Polynomial Functions
- LESSON 6: Factoring GCF and Grouping
- LESSON 7: Factoring Trinomials
- LESSON 8: Special Factoring Situations
- LESSON 9: Polynomial Quiz and Factoring Puzzle
- LESSON 10: Factoring Completely
- LESSON 11: Modeling Equations and Functions with Factoring
- LESSON 12: Polynomial Inequalities
- LESSON 13: Polynomial Review Day 1
- LESSON 14: Polynomial Review Day 2
- LESSON 15: Polynomial Test