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

ABSOLUTEVALUEEQUATIONSANDINEQUALITIES

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

Key Vocabulary: Absolute Value Equations Absolute Value Inequalities


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


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 .


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

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



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



Check:

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


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


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


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


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



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


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


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 .


B. ABSOLUTE VALUE INEQUALITIES: If , then and .

It also applies for : If , then and .


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

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



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



Check:

≮

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



Check:

≮ ≮

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



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


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


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


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



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



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



Check:

≮ ≮

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



Check:

≮ ≮

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


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 .


C. ABSOLUTE VALUE INEQUALITIES: If , then or .

It also applies for : If , then or .



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


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



Check:

≱

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



Check:

≱

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



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



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


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


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



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



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



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



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


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)


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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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