WEEK 10-SOLVING-AND-GRAPHING-LINEAR-INEQUALITIES

Objectives

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Solving and Graphing Linear Inequalities

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2

Introduction

Linear Inequalities

Compound Inequalities

Summary

Two Step Linear Inequalities

Introduction

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Linear Inequalities

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Definition: A linear inequality is a statement in any of these basic forms:

ax+b>0,ax+b≥0,ax+b<0,ax+b≤0.

🗨️

An inequality shows the relationship between two expressions that are not equal.

1

is a variable

are any three numbers.

a,b

Here:

is a variable

x

Remember, each symbol means: > Greater than ≥ Greater than or equals to < Less than ≤ Less than or equals to

a<b, b<cthena<c

If a<bthena+c<b+c

If a<b,c>0thenac<bc

If a<bthena-c<b-c

Let a, b, c be real numbers, the following properties hold:

Properties of Inequalities

If a<b,c<0:

↺

If a<b, c>0thenac>bc

Properties of <,>,

If a<b,c>0:

4. Multiplication Property (Multiplying by a negative number)

Examples If x=3 then x-2=3-2 or: If $3=💵💵💵 Then: $3-$1=💵💵💵-$1

7. Division Property (Dividing by a negative number)

Examples

6. Division Property (Dividing by a positive number)

Example If ❤=4 then 3❤=3×4 3❤=12

Examples x=1 and 1=z then x=z ■=$5 and $5=💵 then ■=💵

3. Subtraction Property

Examples x=1 then 1=x Read as x equals 1 is the same as 2 equals x. Dubious, isn't it? a=-2 then -2=x

2. Addition Property

Examples 1=1 2=2 √2=√2

1. Transitive Property number1<number2 and number2<number3 then number1 is also less than to number3

Examples: Examples 1=1 then 1+2=1+2 x=1 then x+4=1+4

4. Multiplication Property (Multiplying by a positive number)

Try it by yourself:

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Two Step Linear Inequalities

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Let's see some examples:

Example 1: Two step inequality. Solve and plot the solution set.

2x + 1 < 5

x<2

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1. Subtract 1 in both sides.

2. Divide by 2 in both sides.

4. Plot in a number line, take in mind: * x is less than two -> values are on the left of two. * Inequality is "<" so, put number 2 with a ○ indicating number 2 isn't included.

Let's see some examples:

Example 1: Two step inequality. Solve and plot the solution set.

2x-1≥4

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1. Add 1 in both sides.

2. Divide in both sides by 2.

3. Graph the result on a number line. As long as inequality is in the form "≥" In 5/2, there is ●, indicating that this value is included as a possible solution.

Let's see some examples:

Example 2: Solve

Example 3: Three times a number added to 10 equals 25. Find the number.

x=5

2x-1≥4

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Problem might be solved drawing the total (light orange) and subtracting the additional (10).

1. Add 1 in both sides.

2. Divide in both sides by 2.

3. Graph the result on a number line. As long as inequality is in the form "≥" In 5/2, there is ●, indicating that this value is included as a possible solution.

1.

2.

3.

4.

Compound Inequalities

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🗨️

Compound inequalities occur when two inequalities are satisfied simultaneously.

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1≤x≤4

Example 1: A twice a number x added with 1 is between 3 and 9. Find the solution set, graph it and list some possible solutions.

x=2, 2.5, 3 and 4 are some possible solutions.

1. State the problem mathematically: Twice a number x: 2x added with 1: 2x+1 is between 3 and 9: 3≤2x+1≤9

2.Subtract 1 in both sides.

5. List any solution between 1 and 4, take in mind that solutions not only have to be integer.

3. Divide by 2 in both sides.

4. Plot in a number line, take in mind: * x is between 1 and 4 -> Draw a line between 1 and 4. * Inequalities are "≤" so, extreme values have a ●.

Try it by yourself:

Try it by yourself:

Summary:Solving Inequalities

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Additiona+c<b+c

Multiplicationc>0: ac<bcc<0: ac>bc

Subtractiona-c<b-c

Divisionc>0: a/c<b/cc<0: a/c>b/c

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Welcome 6th graders!

A journey soon begin through Social Science experiences!

Great job!

See you next time

8TH-SOLVING-AND-GRAPHING-LINEAR-INEQUALITIES-EN © 2024 by CASURID is licensed under CC BY-NC-ND 4.0

It is highly advised to have:

MATERIAL

• Grid paper.
• Pencils of different colors.
• Eraser.
• A rule.
• A compass.
• A Protactor.
• A calculator.
• Geogebra installed on your phone/tablet/computer (or use online version).

"MA.8.AR.2 Solve multi-step one-variable equations and inequalities."MA.8.AR.2.2 Solve two-step linear inequalities in one variable and represent solutions algebraically and graphically.MA.K12.MTR.6.1 Assess the reasonableness of solutions.MA.K12.MTR.2.1 "Demonstrate understanding by representing problems in multipleways."

Using properties of inequalities

Solve inequality (ies)

It's always a good a idea to visualize possible solutions in a diagram

Plot the solution on a number line

Whether it makes sense or no. Checking validity for some values.

Check the Solution

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State restrictions/conditions in the form of inequalities

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