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Algebra I

Solving & Graphing Inequalities

2016-01-11

www.njctl.org

Slide 3 / 182

Table of Contents

Simple Inequalities Addition/Subtraction

Simple Inequalities Multiplication/Division

Solving Compound Inequalities

Special Cases of Compound Inequalities

Graphing Linear Inequalities in Slope-Intercept Form

click on the topic to go to that section

Glossary & Standards

Solving Systems of Inequalitites

Two-Step and Multiple-Step Inequalities

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Simple Inequalities Involving Additionand Subtraction

Return to Table of Contents

Slide 5 / 182

An Inequality is a mathematical sentence that uses symbols, such as or to compare to quantities.

Inequality

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What do these symbols mean?

LessThan

Less Than or Equal To

Greater Than

Greater Thanor Equal To

click

click

(when read from LEFT to RIGHT)

http://www.njctl.orghttp://www.njctl.orgpage41svgpage24svgpage122svgpage115svgpage167svgpage1svgpage34svgpage65svgpage134svgpage3svg

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Write an inequality for the sentence below:

The sum of a number, n, and fifteen is greater than or equal to nine.

Three times a number, n, is less than 210.

Inequality

Click

Click

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Remember!

Open circle means that number is not included in the solution set and is used to represent < or >.

Closed circle means the solution set includes that number and is used to represent or .

Graphing Inequalities

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Solving one-step inequalities is much like solving one-step equations.

To solve an inequality, you need to isolate the variable using the properties of inequalities and inverse operations.

Solving Inequalities

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To find the solution, isolate the variable x.Remember, it is isolated when it appears by itself on one side of the equation.

Isolate the Variable

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10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

Step 2: Decide whether or not the circle on your boundary should be open or closed based on the symbol used. You can check the computation by substituting the end point of 6 for x. In this case, the end point is not included (open circle) since x < 6.

Solving Inequalities

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Review of Solving Inequalities Using Addition and Subtraction

The following formative assessment questions are review from 7th grade. If further instruction is need, see the presentation at:

http://www.njctl.org/courses/math/7th-grade/equations-inequalities-7th-grade/

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0 1 2 3 4 5-1-2-3-4-5

2 5 6

0 1 2 3 4 5-1-2-3-4-5

2 5 6

0 1 2 3 4 5-1-2-3-4-5

2 5 6

0 1 2 3 4 5-1-2-3-4-5

5 62

A

B

C

D

1 Which graph is the solution to the inequality: a number, n, minus is greater than one third?

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10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

A

B

C

D

2 Which graph is the solution to the inequality ?

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10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10A

B

C

D

3 Which graph is the solution to the inequality ?

http://www.njctl.org/courses/math/7th-grade/equations-inequalities-7th-grade/

Slide 19 / 182

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10A

B

C

D

4 Which graph is the solution to the inequality ?

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10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

1.510 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

1.510 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

1.510 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

1.5A

B

C

D

5 Which graph is the solution to the inequality ?

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Simple Inequalities Involving Multiplication

and Division

Return to Table of Contents

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Again, similarly to solving equations, we can use the properties of multiplication and division to solve and graph inequalities - with one minor difference, which we will encounter in the upcoming slides.

Inequalities Involving Multiplication and Division

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Since x is multiplied by 3, divide both sides by 3 to isolate the variable.

Multiplying or Dividing by a Positive Number

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

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page134svg

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Review of Solving Inequalities Using Multiplication and Division

The following formative assessment questions are review from 7th grade. If further instruction is need, see the presentation at:

http://www.njctl.org/courses/math/7th-grade/equations-inequalities-7th-grade/

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10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

6 Which graph is the solution to the inequality, the product of 4 and a number, x, is greater than 24?

A

B

C

D

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Find the solution to the inequality.9

A

B

C

D

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10

A

B

C

Find the solution to the inequality.

D

http://www.njctl.org/courses/math/7th-grade/equations-inequalities-7th-grade/

Slide 31 / 182

So far, all the operations we have used worked the same as solving equations. The difference between solving

equations versus inequalities is revealed when multiplying or dividing by a negative number.

The direction of the inequality changes only if the number you are using to multiply or divide by is

negative.

Multiplying or Dividing by a Negative Number

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10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

*Note: Dividing each side by -3 changes the to .

Solve and Graph

click for answer

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

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11 Solve the inequality and graph the solution.

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

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12

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

Solve the inequality and graph the solution.

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13

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

Solve the inequality and graph the solution.

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14

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

Solve the inequality and graph the solution.

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In review, an inequality symbol stays the same direction when you:

Add, subtract, multiply or divide by the same positive number on both sides.

Add or subtract the same negative number on both sides.

An inequality symbol changes direction when you:

Multiply or divide by the same negative number on both sides.

Summary

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Solving Two-Step and Multiple-Step

InequalitiesReturn to Table of Contents

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Now we'll solve more complicated inequalities that have multi-step solutions because they involve more than one operation.

Solving inequalities is like solving a puzzle. Keep working through the steps until you get the variable you're looking for alone on one side of the inequality using the same strategies as solving an equation.

Inequalities

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Another reminder! If you multiply or divide by a negative number, reverse the

direction of the inequality symbol!

Multiplying or Dividing by a Negative Number

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page134svg

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10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

Add 9 to both sides

Divide both sides by 4(sign stays the same)

Example: Solve the inequality and graph the solution.

Two Step Inequalities

click for answer

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

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Try these.Solve each inequality and graph each solution.

1.

2.

Solve and Graph

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

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Try these. Solve each inequality and graph the solution.

3.

4.

Solve and Graph

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

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15

A

B

C

Solve and graph the solution.

D

0 1 2 3 4 5-1-2-3-4-5

2.5

0 1 2 3 4 5-1-2-3-4-5

2.5

0 1 2 3 4 5-1-2-3-4-5

2.5

0 1 2 3 4 5-1-2-3-4-5

2.5

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19

A

B

C

Solve and graph the solution.

D

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

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20 Which graph represents the solution set for:

A

B