Solving absolute value equations graphically in unit

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Unformatted text preview: ; −2 3 -3 0 -3 3 x PRE-CALCULUS 11 Unit 6 – Day 3: ABSOLUTE VALUE EQUATIONS ABSOLUTE VALUE EQUATIONS An absolute value equation is an equation that includes an absolute value of a variable expression. SOLVING ABSOLUTE VALUE EQUATIONS GRAPHICALLY In Unit 3, the system y = x2 and y = x + 2 could be solved graphically by finding the coordinates of the points on both graphs. These x-coordinates could also be determined algebraically by using substitution to get the equation x2 = x + 2 ; the roots of this equation will be the x-coordinates of the solutions. Therefore, the roots of the equation x2 = x + 2 can be found by graphing "y = the left side" and "y = the right side" and finding the x-coordinate of the points on both graphs. This graphical method can be used to solve any equation in one variable. example: Solve | 2x − 3 | = 5 graphically. o Graph y = | 2x − 3 | and y y=5 4 y=5 2 o (−1,5) and (4,5) are points on both graphs. Their x-coordinates are −1 and 4. y = |2x − 3| -2 0 x 2 Answer: x = −1 or x = 4 ; solution set = { −1 , 4 } y exercise: Solve | 2x − 3 | = x graphically. 3 -3 0 3 x -3 [Answer: 1 or 3] Unit 6: Day 3 notes - Absolute Value Equations Page 2 of 4 SOLVING ABSOLUTE VALUE EQUATIONS ALGEBRAICALLY To solve absolute value equations algebraically, use the mathematical definition of absolute value: x if x > 0 x if x ≥ 0 | x | = 0 if x = 0 or |x| = − x if x < 0 x if x < 0 example: Solve | 2x − 3 | = 5 algebraically. o | 2x − 3 | is 2x − 3 when 2x − 3 ≥ 0 . Therefore, when x≥ 3 2 2x − 3 = 5 the equation will become Solve this equation. x= 4 This root is the result from an equation that is valid only if x ≥ 3 2 ; 4 should be an acceptable root of the absolute value equation. Check. o | 2x − 3 | is −(2x − 3) when 2x − 3 < 0 . Therefore when the equation will become x< 3 2 −(2x − 3) = 5 x = −1 Solve this equation. This root is the result from an equation that is valid only if x < 3...
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