Simple Inequalities Involving Addition and Subtraction - Inequalities Addition/Subtraction Simple Inequalities Multiplication/Division Solving Compound Inequalities Special Cases of

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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

    Slide 4 / 182

    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/

    Slide 16 / 182

    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?

    Slide 17 / 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 10-1-2-3-4-5-6-7-8-9-10

    A

    B

    C

    D

    2 Which graph is the solution to the inequality ?

    Slide 18 / 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

    3 Which graph is the solution to the inequality ?

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

    B

    C

    D

    4 Which graph is the solution to the inequality ?

    Slide 20 / 182

    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

    Slide 22 / 182

    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

    Slide 24 / 182

    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/

    Slide 26 / 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 10-1-2-3-4-5-6-7-8-9-10

    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

    Slide 27 / 182 Slide 28 / 182

    Slide 29 / 182

    Find the solution to the inequality.9

    A

    B

    C

    D

    Slide 30 / 182

    10

    A

    B

    C

    Find the solution to the inequality.

    D

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

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    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

    Slide 40 / 182

    Slide 41 / 182

    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

    Slide 45 / 182

    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

    Slide 46 / 182

    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

    Slide 47 / 182 Slide 48 / 182

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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

    Slide 51 / 182

    20 Which graph represents the solution set for:

    A

    B

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