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Unformatted text preview: ; −2 3 3 0 3 3 x PRECALCULUS 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 xcoordinates could also be determined
algebraically by using substitution to get the equation x2 = x + 2 ; the roots of this
equation will be the xcoordinates 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 xcoordinate 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 xcoordinates 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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 Fall '11
 Aytona
 Calculus, PreCalculus

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