– 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 because both 8 and -8 are located a distance of 8 from 0.

Absolute Value (of x) Symbol lxl The distance x is from 0 on the number line. Always positive Ex: l-3l=3 -4 -3 -2 -1 0 1 2 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.

Solving an Absolute-Value Equation: Solve | x  2 |  5 Solve | 2x  7 |  5  4

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

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

Solve the following Absolute-Value Equation: Practice: 1) Solve 6x-3 = 15 2) Solve 2x + 7 -3 = 8

* Plug in answers to check your solutions! 1) Solve 6x-3 = 15 6x-3 = 15 or 6x-3 = -15 6x = 18 or 6x = -12 x = 3 or x = -2 * Plug in answers to check your solutions!

Get the abs. value part by itself first! Answer :: 2) Solve 2x + 7 -3 = 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.

***Important NOTE*** 3 2x + 9 +12 = 10 - 12 - 12 3 2x + 9 = - 2 3 3 - 12 - 12 3 2x + 9 = - 2 3 3 No Solution 2x + 9 = - 2 3 What about this absolute value equation? 3x – 6 – 5 = – 7

Solving & Graphing Absolute Value Inequalities

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.

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 + 9 + 9 + 9 + 9 4x ≤ 30 4x ≥ -12 4 4 4 4 x ≤ 7.5 x ≥ -3 -3 7 8

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 > 7 or 2x + 1 < - 7 – 1 - 1 – 1 - 1 2x > 6 2x < - 8 2 2 2 2 x < - 4 x > 3 3 -4

Solving Absolute Value Inequalities: Solve | x  4 | < 3 and graph the solution. Solve | 2x  1 | 3  6 and graph the solution.

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.

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  1 0 1 2 3 4 5 6

Solve and graph the following Absolute-Value Inequalities: 3) |x -5| < 3

Answer :: Solve & graph. 3) Get absolute value by itself first. Becomes an “or” problem -2 3 4

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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