To solve absolute value equations algebraically, you'll isolate the absolute value expression, then consider both the positive and negative possibilities of the expression inside the absolute value.
Here's a step-by-step guide:
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Isolate the Absolute Value Expression: Get the absolute value expression by itself on one side of the equation. This might involve adding, subtracting, multiplying, or dividing terms on both sides.
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Set Up Two Equations: Once the absolute value is isolated, create two separate equations:
- One equation where the expression inside the absolute value is equal to the positive value on the other side of the equation.
- Another equation where the expression inside the absolute value is equal to the negative value on the other side of the equation.
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Solve Each Equation: Solve each of the two equations you created in step 2. These are now standard algebraic equations.
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Check Your Solutions: Substitute each solution back into the original absolute value equation to verify that it is a valid solution. Absolute value equations can sometimes produce extraneous solutions (solutions that satisfy the derived equations but not the original equation).
Example:
Solve the equation: |x - 1| = 3
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Isolate: The absolute value expression is already isolated.
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Two Equations:
- x - 1 = 3
- x - 1 = -3
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Solve:
- x - 1 = 3 => x = 4
- x - 1 = -3 => x = -2
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Check:
- |4 - 1| = |3| = 3 (Valid)
- |-2 - 1| = |-3| = 3 (Valid)
Therefore, the solutions are x = 4 and x = -2.
In Summary: Absolute value equations require you to consider both positive and negative possibilities to find all valid solutions. Always check your answers by plugging them back into the original equation.
