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Absolute Value Equations and Inequalities Unit 3 Lesson 7
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Absolute Value Equations and Inequalities · ABSOLUTE VALUE EQUATIONS AND INEQUALITIES B. ABSOLUTE VALUE INEQUALITIES: O When solving inequalities of the form O , we need to find

Jun 23, 2020

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Page 1: Absolute Value Equations and Inequalities · ABSOLUTE VALUE EQUATIONS AND INEQUALITIES B. ABSOLUTE VALUE INEQUALITIES: O When solving inequalities of the form O , we need to find

Absolute Value Equations and InequalitiesUnit 3 Lesson 7

Page 2: Absolute Value Equations and Inequalities · ABSOLUTE VALUE EQUATIONS AND INEQUALITIES B. ABSOLUTE VALUE INEQUALITIES: O When solving inequalities of the form O , we need to find

ABSOLUTEVALUEEQUATIONSANDINEQUALITIES

Students will be able to:solve absolute value equations and inequalities.

Key Vocabulary: Absolute Value Equations Absolute Value Inequalities

Page 3: Absolute Value Equations and Inequalities · ABSOLUTE VALUE EQUATIONS AND INEQUALITIES B. ABSOLUTE VALUE INEQUALITIES: O When solving inequalities of the form O , we need to find

ABSOLUTEVALUEEQUATIONSANDINEQUALITIES

The three types of open sentences that can involve absolute value:

Page 4: Absolute Value Equations and Inequalities · ABSOLUTE VALUE EQUATIONS AND INEQUALITIES B. ABSOLUTE VALUE INEQUALITIES: O When solving inequalities of the form O , we need to find

ABSOLUTEVALUEEQUATIONSANDINEQUALITIES

A.  ABSOLUTE VALUE EQUATIONS: When solving equations involving absolute value, we need to consider these cases

a.  The value inside the absolute value symbols is positive. 

b.  The value inside the absolute value symbols is negative.

If , then  or  .

Page 5: Absolute Value Equations and Inequalities · ABSOLUTE VALUE EQUATIONS AND INEQUALITIES B. ABSOLUTE VALUE INEQUALITIES: O When solving inequalities of the form O , we need to find

ABSOLUTEVALUEEQUATIONSANDINEQUALITIES

Sample Problem 1: Solve each equation and graph the solution set.A. | |

0 1 2 3 4-1-2 5 6 7 8 9

Page 6: Absolute Value Equations and Inequalities · ABSOLUTE VALUE EQUATIONS AND INEQUALITIES B. ABSOLUTE VALUE INEQUALITIES: O When solving inequalities of the form O , we need to find

ABSOLUTEVALUEEQUATIONSANDINEQUALITIES

Sample Problem 1: Solve each equation and graph the solution set.A. | |

0 1 2 3 4-1-2 5 6 7 8 9

Page 7: Absolute Value Equations and Inequalities · ABSOLUTE VALUE EQUATIONS AND INEQUALITIES B. ABSOLUTE VALUE INEQUALITIES: O When solving inequalities of the form O , we need to find

ABSOLUTEVALUEEQUATIONSANDINEQUALITIES

Sample Problem 1: Solve each equation and graph the solution set.A. | |

Check:

0 1 2 3 4-1-2 5 6 7 8 9

Page 8: Absolute Value Equations and Inequalities · ABSOLUTE VALUE EQUATIONS AND INEQUALITIES B. ABSOLUTE VALUE INEQUALITIES: O When solving inequalities of the form O , we need to find

ABSOLUTEVALUEEQUATIONSANDINEQUALITIES

Sample Problem 1: Solve each equation and graph the solution set.B.

0 2 4 6 8-2-4-6-8 10 12-10-12

Page 9: Absolute Value Equations and Inequalities · ABSOLUTE VALUE EQUATIONS AND INEQUALITIES B. ABSOLUTE VALUE INEQUALITIES: O When solving inequalities of the form O , we need to find

ABSOLUTEVALUEEQUATIONSANDINEQUALITIES

Sample Problem 1: Solve each equation and graph the solution set.B. /

/

0 2 4 6 8-2-4-6-8 10 12-10-12

Page 10: Absolute Value Equations and Inequalities · ABSOLUTE VALUE EQUATIONS AND INEQUALITIES B. ABSOLUTE VALUE INEQUALITIES: O When solving inequalities of the form O , we need to find

ABSOLUTEVALUEEQUATIONSANDINEQUALITIES

Sample Problem 1: Solve each equation and graph the solution set.B. / Check: //

0 2 4 6 8-2-4-6-8 10 12-10-12

Page 11: Absolute Value Equations and Inequalities · ABSOLUTE VALUE EQUATIONS AND INEQUALITIES B. ABSOLUTE VALUE INEQUALITIES: O When solving inequalities of the form O , we need to find

ABSOLUTEVALUEEQUATIONSANDINEQUALITIES

Sample Problem 1: Solve each equation and graph the solution set.C.

0 2 4 6 8-2-4-6-8 10 12-10-12

Page 12: Absolute Value Equations and Inequalities · ABSOLUTE VALUE EQUATIONS AND INEQUALITIES B. ABSOLUTE VALUE INEQUALITIES: O When solving inequalities of the form O , we need to find

ABSOLUTEVALUEEQUATIONSANDINEQUALITIES

Sample Problem 1: Solve each equation and graph the solution set.C.

0 2 4 6 8-2-4-6-8 10 12-10-12

Page 13: Absolute Value Equations and Inequalities · ABSOLUTE VALUE EQUATIONS AND INEQUALITIES B. ABSOLUTE VALUE INEQUALITIES: O When solving inequalities of the form O , we need to find

ABSOLUTEVALUEEQUATIONSANDINEQUALITIES

Sample Problem 1: Solve each equation and graph the solution set.C. Check:

0 2 4 6 8-2-4-6-8 10 12-10-12

Page 14: Absolute Value Equations and Inequalities · ABSOLUTE VALUE EQUATIONS AND INEQUALITIES B. ABSOLUTE VALUE INEQUALITIES: O When solving inequalities of the form O , we need to find

ABSOLUTEVALUEEQUATIONSANDINEQUALITIES

B. ABSOLUTE VALUE INEQUALITIES: When solving inequalities of the form , we need to find the Intersection of these cases: 

a.  The value inside the absolute value symbols is less than the positive value of . 

b.  The value inside the absolute value symbols is greater than the negative value of . 

Page 15: Absolute Value Equations and Inequalities · ABSOLUTE VALUE EQUATIONS AND INEQUALITIES B. ABSOLUTE VALUE INEQUALITIES: O When solving inequalities of the form O , we need to find

ABSOLUTEVALUEEQUATIONSANDINEQUALITIES

B. ABSOLUTE VALUE INEQUALITIES: If  , then  and  .

It also applies for  : If  , then  and  .

Page 16: Absolute Value Equations and Inequalities · ABSOLUTE VALUE EQUATIONS AND INEQUALITIES B. ABSOLUTE VALUE INEQUALITIES: O When solving inequalities of the form O , we need to find

ABSOLUTEVALUEEQUATIONSANDINEQUALITIES

Sample Problem 2: Solve each equation and graph the solution set.A.

0 1 2 3 4-1-2 5 6 7 8 9

Page 17: Absolute Value Equations and Inequalities · ABSOLUTE VALUE EQUATIONS AND INEQUALITIES B. ABSOLUTE VALUE INEQUALITIES: O When solving inequalities of the form O , we need to find

ABSOLUTEVALUEEQUATIONSANDINEQUALITIES

Sample Problem 2: Solve each equation and graph the solution set.A.

0 1 2 3 4-1-2 5 6 7 8 9

Page 18: Absolute Value Equations and Inequalities · ABSOLUTE VALUE EQUATIONS AND INEQUALITIES B. ABSOLUTE VALUE INEQUALITIES: O When solving inequalities of the form O , we need to find

ABSOLUTEVALUEEQUATIONSANDINEQUALITIES

Sample Problem 2: Solve each equation and graph the solution set.A.

Check:

≮

0 1 2 3 4-1-2 5 6 7 8 9

Page 19: Absolute Value Equations and Inequalities · ABSOLUTE VALUE EQUATIONS AND INEQUALITIES B. ABSOLUTE VALUE INEQUALITIES: O When solving inequalities of the form O , we need to find

ABSOLUTEVALUEEQUATIONSANDINEQUALITIES

Sample Problem 2: Solve each equation and graph the solution set.A.

Check:

≮ ≮

0 1 2 3 4-1-2 5 6 7 8 9

Page 20: Absolute Value Equations and Inequalities · ABSOLUTE VALUE EQUATIONS AND INEQUALITIES B. ABSOLUTE VALUE INEQUALITIES: O When solving inequalities of the form O , we need to find

ABSOLUTEVALUEEQUATIONSANDINEQUALITIES

Sample Problem 2: Solve each equation and graph the solution set.B.

0 2 4 6 8-2-4-6-8 10-14 -10-12

Page 21: Absolute Value Equations and Inequalities · ABSOLUTE VALUE EQUATIONS AND INEQUALITIES B. ABSOLUTE VALUE INEQUALITIES: O When solving inequalities of the form O , we need to find

ABSOLUTEVALUEEQUATIONSANDINEQUALITIES

Sample Problem 2: Solve each equation and graph the solution set.B. /

/

0 2 4 6 8-2-4-6-8 10-14 -10-12

Page 22: Absolute Value Equations and Inequalities · ABSOLUTE VALUE EQUATIONS AND INEQUALITIES B. ABSOLUTE VALUE INEQUALITIES: O When solving inequalities of the form O , we need to find

ABSOLUTEVALUEEQUATIONSANDINEQUALITIES

Sample Problem 2: Solve each equation and graph the solution set.B. /Check: / //

≰

0 2 4 6 8-2-4-6-8 10-14 -10-12

Page 23: Absolute Value Equations and Inequalities · ABSOLUTE VALUE EQUATIONS AND INEQUALITIES B. ABSOLUTE VALUE INEQUALITIES: O When solving inequalities of the form O , we need to find

ABSOLUTEVALUEEQUATIONSANDINEQUALITIES

Sample Problem 2: Solve each equation and graph the solution set.B. /Check:

≰

0 2 4 6 8-2-4-6-8 10-14 -10-12

Page 24: Absolute Value Equations and Inequalities · ABSOLUTE VALUE EQUATIONS AND INEQUALITIES B. ABSOLUTE VALUE INEQUALITIES: O When solving inequalities of the form O , we need to find

ABSOLUTEVALUEEQUATIONSANDINEQUALITIES

Sample Problem 2: Solve each equation and graph the solution set.C.

0 2 4 6 8-2-4-6-8 10 12-10-12

Page 25: Absolute Value Equations and Inequalities · ABSOLUTE VALUE EQUATIONS AND INEQUALITIES B. ABSOLUTE VALUE INEQUALITIES: O When solving inequalities of the form O , we need to find

ABSOLUTEVALUEEQUATIONSANDINEQUALITIES

Sample Problem 2: Solve each equation and graph the solution set.C.

0 2 4 6 8-2-4-6-8 10 12-10-12

Page 26: Absolute Value Equations and Inequalities · ABSOLUTE VALUE EQUATIONS AND INEQUALITIES B. ABSOLUTE VALUE INEQUALITIES: O When solving inequalities of the form O , we need to find

ABSOLUTEVALUEEQUATIONSANDINEQUALITIES

Sample Problem 2: Solve each equation and graph the solution set.C.

Check:

≮ ≮

0 2 4 6 8-2-4-6-8 10 12-10-12

Page 27: Absolute Value Equations and Inequalities · ABSOLUTE VALUE EQUATIONS AND INEQUALITIES B. ABSOLUTE VALUE INEQUALITIES: O When solving inequalities of the form O , we need to find

ABSOLUTEVALUEEQUATIONSANDINEQUALITIES

Sample Problem 2: Solve each equation and graph the solution set.C.

Check:

≮ ≮

0 2 4 6 8-2-4-6-8 10 12-10-12

Page 28: Absolute Value Equations and Inequalities · ABSOLUTE VALUE EQUATIONS AND INEQUALITIES B. ABSOLUTE VALUE INEQUALITIES: O When solving inequalities of the form O , we need to find

ABSOLUTEVALUEEQUATIONSANDINEQUALITIES

C. ABSOLUTE VALUE INEQUALITIES: When solving inequalities of the form , we need to find the Intersection of these cases: 

a. The value inside the absolute value symbols is GREATER THAN THE POSITIVE VALUE OF . 

b. The value inside the absolute value symbols is LESS THAN THE NEGATIVE VALUE OF . 

Page 29: Absolute Value Equations and Inequalities · ABSOLUTE VALUE EQUATIONS AND INEQUALITIES B. ABSOLUTE VALUE INEQUALITIES: O When solving inequalities of the form O , we need to find

ABSOLUTEVALUEEQUATIONSANDINEQUALITIES

C. ABSOLUTE VALUE INEQUALITIES: If  , then  or  .

It also applies for : If  , then  or  .

Page 30: Absolute Value Equations and Inequalities · ABSOLUTE VALUE EQUATIONS AND INEQUALITIES B. ABSOLUTE VALUE INEQUALITIES: O When solving inequalities of the form O , we need to find

ABSOLUTEVALUEEQUATIONSANDINEQUALITIES

Sample Problem 3: Solve each equation and graph the solution set.A.

0 4 8 12 16-4-8-12-16 20 24-20-24

Page 31: Absolute Value Equations and Inequalities · ABSOLUTE VALUE EQUATIONS AND INEQUALITIES B. ABSOLUTE VALUE INEQUALITIES: O When solving inequalities of the form O , we need to find

ABSOLUTEVALUEEQUATIONSANDINEQUALITIES

Sample Problem 3: Solve each equation and graph the solution set.A. or   

0 4 8 12 16-4-8-12-16 20 24-20-24

Page 32: Absolute Value Equations and Inequalities · ABSOLUTE VALUE EQUATIONS AND INEQUALITIES B. ABSOLUTE VALUE INEQUALITIES: O When solving inequalities of the form O , we need to find

ABSOLUTEVALUEEQUATIONSANDINEQUALITIES

Sample Problem 3: Solve each equation and graph the solution set.A.    or   

Check:

≱

0 4 8 12 16-4-8-12-16 20 24-20-24

Page 33: Absolute Value Equations and Inequalities · ABSOLUTE VALUE EQUATIONS AND INEQUALITIES B. ABSOLUTE VALUE INEQUALITIES: O When solving inequalities of the form O , we need to find

ABSOLUTEVALUEEQUATIONSANDINEQUALITIES

Sample Problem 3: Solve each equation and graph the solution set.A.    or   

Check:

≱

0 4 8 12 16-4-8-12-16 20 24-20-24

Page 34: Absolute Value Equations and Inequalities · ABSOLUTE VALUE EQUATIONS AND INEQUALITIES B. ABSOLUTE VALUE INEQUALITIES: O When solving inequalities of the form O , we need to find

ABSOLUTEVALUEEQUATIONSANDINEQUALITIES

Sample Problem 3: Solve each equation and graph the solution set.B.

0 4 8 12 16-4-8-12-16 20 24-20-24

Page 35: Absolute Value Equations and Inequalities · ABSOLUTE VALUE EQUATIONS AND INEQUALITIES B. ABSOLUTE VALUE INEQUALITIES: O When solving inequalities of the form O , we need to find

ABSOLUTEVALUEEQUATIONSANDINEQUALITIES

Sample Problem 3: Solve each equation and graph the solution set.B.

0 4 8 12 16-4-8-12-16 20 24-20-24

Page 36: Absolute Value Equations and Inequalities · ABSOLUTE VALUE EQUATIONS AND INEQUALITIES B. ABSOLUTE VALUE INEQUALITIES: O When solving inequalities of the form O , we need to find

ABSOLUTEVALUEEQUATIONSANDINEQUALITIES

Sample Problem 3: Solve each equation and graph the solution set.B. Check:

≯ ≯

0 4 8 12 16-4-8-12-16 20 24-20-24

Page 37: Absolute Value Equations and Inequalities · ABSOLUTE VALUE EQUATIONS AND INEQUALITIES B. ABSOLUTE VALUE INEQUALITIES: O When solving inequalities of the form O , we need to find

ABSOLUTEVALUEEQUATIONSANDINEQUALITIES

Sample Problem 3: Solve each equation and graph the solution set.B. Check:

≯ ≯

0 4 8 12 16-4-8-12-16 20 24-20-24

Page 38: Absolute Value Equations and Inequalities · ABSOLUTE VALUE EQUATIONS AND INEQUALITIES B. ABSOLUTE VALUE INEQUALITIES: O When solving inequalities of the form O , we need to find

ABSOLUTEVALUEEQUATIONSANDINEQUALITIES

Sample Problem 3: Solve each equation and graph the solution set.C.

0 4 8 12 16-4-8-12-16 20 24-20-24

Page 39: Absolute Value Equations and Inequalities · ABSOLUTE VALUE EQUATIONS AND INEQUALITIES B. ABSOLUTE VALUE INEQUALITIES: O When solving inequalities of the form O , we need to find

ABSOLUTEVALUEEQUATIONSANDINEQUALITIES

Sample Problem 3: Solve each equation and graph the solution set.C.

0 4 8 12 16-4-8-12-16 20 24-20-24

Page 40: Absolute Value Equations and Inequalities · ABSOLUTE VALUE EQUATIONS AND INEQUALITIES B. ABSOLUTE VALUE INEQUALITIES: O When solving inequalities of the form O , we need to find

ABSOLUTEVALUEEQUATIONSANDINEQUALITIES

Sample Problem 3: Solve each equation and graph the solution set.C. Check:

0 4 8 12 16-4-8-12-16 20 24-20-24

Page 41: Absolute Value Equations and Inequalities · ABSOLUTE VALUE EQUATIONS AND INEQUALITIES B. ABSOLUTE VALUE INEQUALITIES: O When solving inequalities of the form O , we need to find

ABSOLUTEVALUEEQUATIONSANDINEQUALITIES

Sample Problem 3: Solve each equation and graph the solution set.C. Check:

0 4 8 12 16-4-8-12-16 20 24-20-24

Page 42: Absolute Value Equations and Inequalities · ABSOLUTE VALUE EQUATIONS AND INEQUALITIES B. ABSOLUTE VALUE INEQUALITIES: O When solving inequalities of the form O , we need to find

ABSOLUTEVALUEEQUATIONSANDINEQUALITIES

Sample Problem 4: The starting players of the school’s varsity basketball team have an average scoring point between 9 and 20. Write an absolute value inequality describing the average scoring point of the starting players.

Step 1 Write the inequality.

Step 2 Determine the midpoint.

Step 3 Determine the distance from the midpoint.

Step 4 Write the absolute value inequality. (Use appropriate inequality symbol)

Page 43: Absolute Value Equations and Inequalities · ABSOLUTE VALUE EQUATIONS AND INEQUALITIES B. ABSOLUTE VALUE INEQUALITIES: O When solving inequalities of the form O , we need to find

ABSOLUTEVALUEEQUATIONSANDINEQUALITIES

Sample Problem 4: The starting players of the school’s varsity basketball team have an average scoring point between 9 and 20. Write an absolute value inequality describing the average scoring point of the starting players.

Step 1 Write the inequality.

Step 2 Determine the midpoint. .Step 3 Determine the distance from the midpoint. . . . .Step 4 Write the absolute value inequality. (Use appropriate inequality symbol) . .


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