Solve Quadratic Inequalities

2nd Quarter Math 9

by Patricia Diomampo

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​ESLR

Through understanding Quadratic equations, Faithians will acknowledge God as a God of Order and that rules are made out of His love for mankind through His holiness. Faithians will desire to read, study, and obey God’s laws and school rules and regulations.

Bible Verse:

Joshua 1:8

“Keep this Book of the Law always on your lips; meditate on it day and night, so that you may be careful to do everything written in it. Then you will be prosperous and successful”

Principle:

Faithians read, study and obey God’s laws and school rules and regulations.

​Aim:

​I will be able to...

• ​Illustrate quadratic inequalities

• Solve quadratic inequalities.

• Solve problems involving quadratic inequalities.

Review

​1.) What is a quadratic equation?

2.) What is the standard form of a quadratic equation?

3.) Give examples of a quadratic equation.

​Action

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​Analysis

​Suppose we are going to change the equal sign (=) of the quadratic equation  into >, this becomes . Is this still a quadratic equation?

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​Quadratic Inequality.

the different inequality symbols.

>        greater than

<        less than

≥        greater than or equal to

≤        less than or equal to

Multiple Choice

Quadratic Inequality?

15-2x=3x²

Multiple Choice

$$\left(3x^2+8\right)\ +\left(x+6\right)\ge-4$$  

Quadratic Inequality?

Multiple Choice

$$x^2-4<x^2-2x$$  

Quadratic Inequality?

Multiple Choice

$$2+x^2-x<0$$  

Quadratic Inequality?

​Which of the following mathematical sentences are quadratic inequalities?

a.) x² + 9x +20 = 0

b.) 2t² < 21- 9t

c.) r² + 10r ≤ - 16

d.) 3w² + 12w < 0

e.) 2s² + 7s +5 ≥ 0

f.) 15 - 6h² = 10

g.) 4x² - 25 = 0

h.) m² = 6m – 7

​10/25/21

​The city government is planning to construct a new children’s playground. It wants to fence in a rectangular ground using one of the walls of a building. The length of the new playground is 15m longer than its width and its area is greater than the old playground. The area of the old playground is 2200 m².

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​What is the correct inequality or mathematical sentence  to be used on the given problem?

​Answer

​A: w² + 15w > 2200

​Mr. Villegas has a vacant lot in his backyard. He wants to make as many rectangular garden as posible such that the length of each garden is 2 meters longer than its width. He also wants the length of the garden with the smallest área to be 3 meter.

​​A: w² + 2w < 3

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​The floor of a conference hall can be covered completely with tiles. Its length is 36 feet longer than its width. The area of the floor is less than 2040  square feet.

​Generalization/ Summary

·         What is a Quadratic Inequality?

·         When can we tell that a quadratic inequality is in standard form?

​* What are the symbols used in quadratic inequality?

​Give examples of quadratic inequality in standard form

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​10/26/21

Open Ended

What have you learned today?

Poll

How's our lesson today?

10/27/21​

Aim:

​I will be able to...

• Solve quadratic inequalities.

• Solve problems involving quadratic inequalities.

​Action

What have you observed on the inequality used in each number line? What can you conclude?

​11/2/21

​Discussion

​https://www.youtube.com/watch?v=Pw6S7xh_9qU

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​refer to shared screen

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​ Find the solution set of  x² + 7x +12 > 0.

Step 1. Change the inequality to standard form of quadratic equation ax² + bx + c = 0.

​                x² + 7x +12 = 0

Step 2. Find the value of x by using any of the 4 methods in solving quadratic equation.

                     x = - 3,      x = - 4

 

Step 3. Make 3 intervals.

          a.) -∞ < x < -4

          b.) -4 < x < -3

          c.) -3 < x < ∞

 

​Steps in solving Quadratic Inequalities

​Steps in Solving Quadratic Inequalities

​Step 4. Solve to test a number from each interval against the inequality.

​a.) for -∞  < x < - 4,  let x = -5:

       x² + 7x +12 > 0

 -5² + 7(-5) + 12 > 0

        25 -35 +12 > 0

             -10 +12 >0

                       2 > 0   True

b.) for -4 < x < - 3 ,  let x = -3.1:

          x² + 7x +12 > 0

-3.1² + 7(-3.1)+12 > 0

    9.61 – 21.7 +12 > 0

            -31.31 +12 > 0

                   -19.31 > 0 False

​c.) for -3 < x < ∞,  let x = 0

          x² + 7x +12 > 0

        0² + 7(0)+12 > 0

             0 + 0 +12 > 0

                         12 > 0

                           True

​Step 5. Test also the points x = -3, x = -4 by using the given inequality if they too are also part of the solution or not.

a.) let x = -4

      x² + 7x +12 > 0

-4² + 7(-4) + 12 > 0

     16 -28  + 12 > 0

            -12 +12 > 0

                       0 > 0

                       True

​b.) let x = -3

    x² + 7x +12 > 0

-3² + 7(-3)+12 > 0

       9 -21 +12 > 0

           -12+12 > 0

                     0 > 0

                     True

​Step 6. Plot the corresponding points on the number line.  Hollow circles are used in the graph to show that (-3) and (-4) are not part of the solution set. Bold circles if they are part of the solutions.

 

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Step 7. Write the solution set.

            { x : x < -4 or x > -3 }  

​Application

​Individual Activity:

Find the solution set of  x² + 4x +3 ≤ 0.

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​11/4/21

Fill in the Blank

Find the solution set of  x² + 4x +3 ≤ 0. Write the solution using inequality symbol.

Open Ended

What have you learned today?

Poll

How's our lesson today?

​Aim

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Recalls the steps in solving problems involving quadratic inequalities.

Solves problems involving quadratic inequalities.

​Action

​“The perimeter of a rectangle is 24 cm. If x is equivalent to its length, find its possible dimension if the area of the same rectangle is less than 35 sq. cm.”

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​The possible dimension of the area of the same rectangle which is less than 35 sq. cm could be :

L = 8 cm

W = 4 cm

or  8 cm by 4 cm

​Examples:

​The length of a wall is 17 m more than its width. If the area of the wall is less than 60 m², what could be the length of the wall?

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​answer: 18 m

​Example 2:

​A rectangular box is completely filled with dice. Each die has a volume of 1cm³. The length of the box is 3cm greater than its width and its height is 5 cm. Suppose the box holds at most 140 dice. What are the possible dimensions of the box?

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​Answer:

width = 4 cm, length = 7 cm, height = 5 cm

width = 3 cm, length = 6 cm, height = 5 cm

Open Ended

What have you learned today?

Poll

How's our lesson today?

​Aim:

●Differentiates quadratic from linear functions

●Describes quadratic function and its application to real life

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Motivation

​1.    Have you ever asked yourself why PBA star players are good in free throws?

​2.    How do angry bird expert players hit their targets?

​Action and Analysis

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​6. How can you recognize a quadratic function when a table of values is given?

refer to jamboard shared on screen

• ​Solve by completing the square

• ​Solve by applying the formula

​ h= -b/2a k= 4ac-b²/4a

​Application

Fill in the Blank

What is the hidden message?

Type in all caps

Generalization:

By group, describe the ways ​of recognizing a quadratic function

​Individual work:

Open Ended

What have you learned today?

Poll

How's our lesson today?

​Maximum and Minimum Values of a Quadratic Function

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in the vertex form (h, k),

h- determines the axis of symmetry

​k- determines the maximum or minimum of the graph