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Name: Date: Period: Topic: Solving Absolute Value Equations & Inequalities Essential Question: What is the process needed to solve absolute value equations and inequalities? Warm-Up: Describe the similarities and differences between equations and inequalities.

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**Home-Learning #2 Review**

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Quiz #7:

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**– 8 and 8 is a solution of the**

Recall : Absolute value | x | : is the distance between x and 0. If | x | = 8, then – 8 and 8 is a solution of the equation ; or | x | 8, then any number between 8 and 8 is a solution of the inequality.

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**Absolute Value (of x) Symbol lxl**

The distance x is from 0 on the number line. Always positive Ex: l-3l=3 Recall: You can solve some absolute-value equations using mental math. For instance, you learned that the equation | x | 3 has two solutions: 3 and 3. To solve absolute-value equations, you can use the fact that the expression inside the absolute value symbols can be either positive or negative.

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**Solving an Absolute-Value Equation:**

Solve | x 2 | 5 Solve | 2x 7 | 5 4

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**Answer :: Solving an Absolute-Value Equation Solve | x 2 | 5**

The expression x 2 can be equal to 5 or 5. x 2 IS POSITIVE | x 2 | 5 x 2 5 x 7 x 3 x 2 IS NEGATIVE | x 2 | 5 x 2 5 The equation has two solutions: 7 and –3. CHECK | 7 2 | | 5 | 5 | 3 2 | | 5 | 5

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Answer :: Solve | 2x 7 | 5 4 Isolate the absolute value expression on one side of the equation. SOLUTION Isolate the absolute value expression on one side of the equation. 2x 7 IS POSITIVE | 2x 7 | 5 4 | 2x 7 | 9 2x 7 +9 2x 16 2x 7 IS NEGATIVE | 2x 7 | 5 4 | 2x 7 | 9 2x 7 9 2x 2 2x 7 IS POSITIVE 2x 7 IS POSITIVE 2x 7 +9 2x 7 IS NEGATIVE 2x 7 IS NEGATIVE 2x 7 9 | 2x 7 | 5 4 | 2x 7 | 5 4 | 2x 7 | 9 | 2x 7 | 9 2x 7 +9 2x 7 9 2x 16 2x 2 x 8 x 1 TWO SOLUTIONS x 1

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**Solve the following Absolute-Value Equation:**

Practice: 1) Solve 6x-3 = 15 2) Solve 2x = 8

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*** Plug in answers to check your solutions!**

1) Solve 6x-3 = 15 6x-3 = or 6x-3 = -15 6x = or 6x = -12 x = 3 or x = -2 * Plug in answers to check your solutions!

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**Get the abs. value part by itself first!**

Answer :: 2) Solve 2x = 8 Get the abs. value part by itself first! 2x+7 = 11 Now split into 2 parts. 2x+7 = 11 or 2x+7 = -11 2x = 4 or 2x = -18 x = 2 or x = -9 Check the solutions.

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*****Important NOTE*** 3 2x + 9 +12 = 10 - 12 - 12 3 2x + 9 = - 2 3 3**

3 2x = - 2 No Solution 2x = - 2 3 What about this absolute value equation? 3x – 6 – 5 = – 7

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**Solving & Graphing Absolute Value Inequalities**

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**Solving an Absolute Value Inequality:**

Step 1: Rewrite the inequality as a conjunction or a disjunction. If you have a you are working with a conjunction or an ‘and’ statement. Remember: “Less thand” If you have a you are working with a disjunction or an ‘or’ statement. Remember: “Greator” Step 2: In the second equation you must negate the right hand side and reverse the direction of the inequality sign. Solve as a compound inequality.

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**Ex: “and” inequality 4x – 9 ≤ 21 4x – 9 ≥ -21 + 9 + 9 + 9 + 9 4x ≤ 30**

Becomes an “and” problem Positive Negative 4x – 9 ≤ 21 4x – 9 ≥ -21 4x ≤ 30 4x ≥ -12 x ≤ 7.5 x ≥ -3

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**This is an ‘or’ statement. (Greator).**

Ex: “or” inequality In the 2nd inequality, reverse the inequality sign and negate the right side value. |2x + 1| > 7 2x + 1 > or x + 1 < - 7 – – 2x > 6 2x < - 8 x < - 4 x > 3 3 -4

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**Solving Absolute Value Inequalities:**

Solve | x 4 | < 3 and graph the solution. Solve | 2x 1 | 3 6 and graph the solution.

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**Answer :: Solve | x 4 | < 3 Reverse inequality symbol.**

x 4 IS POSITIVE x 4 IS NEGATIVE | x 4 | 3 | x 4 | 3 x 4 3 x 4 3 Reverse inequality symbol. x 7 x 1 The solution is all real numbers greater than 1 and less than 7. This can be written as 1 x 7.

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**Solve | 2x 1 | 3 6 and graph the solution.**

Answer :: Solve | 2x 1 | 3 6 and graph the solution. | 2x 1 | 3 6 | 2x 1 | 9 2x 1 +9 x 4 2x 8 | 2x 1 | 3 6 2x 1 9 2x 10 x 5 2x + 1 IS POSITIVE 2x + 1 IS NEGATIVE Reverse inequality symbol. The solution is all real numbers greater than or equal to 4 or less than or equal to 5. This can be written as the compound inequality x 5 or x 4. 6 5 4 3 2

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**Solve and graph the following Absolute-Value Inequalities:**

3) |x -5| < 3

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**Answer :: Solve & graph. 3) Get absolute value by itself first.**

Becomes an “or” problem

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**Answer :: |x -5|< 3 x -5< 3 and x -5< 3**

This is an ‘and’ statement. (Less thand). Rewrite. In the 2nd inequality, reverse the inequality sign and negate the right side value. Solve each inequality. Graph the solution. |x -5|< 3 x -5< 3 and x -5< 3 x -5< 3 and x -5> -3 x < 8 and x > 2 2 < x < 8 8 2

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**Solve and Graph 5) 4m - 5 > 7 or 4m - 5 < - 9**

6) 3 < x - 2 < 7 7) |y – 3| > 1 |p + 2| + 4 < 10 |3t - 2| + 6 = 2

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Home-Learning #3: Page (18, 26,36, 40, 64)

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