Math Problem Statement
=-6 solve for both equations check for extraneous solutions
Solution
We are given the equation:
Step 1: Eliminate the denominator by multiplying both sides by -2.
Step 2: Set up two equations based on the absolute value definition.
The absolute value equation gives us two possibilities:
Step 3: Solve both equations.
For the first equation:
For the second equation:
Step 4: Check for extraneous solutions.
We must substitute both solutions into the original equation to verify.
Checking :
This is correct.
Checking :
This is also correct.
Conclusion:
Both and are valid solutions, and there are no extraneous solutions.
Would you like further details on this? Here are 5 related questions:
- How does the absolute value affect solving equations?
- What is an extraneous solution, and why do we check for it?
- Can an absolute value equation ever have no solution?
- How would this process change if the equation was a fraction with a positive denominator?
- Can absolute value equations have more than two solutions?
Tip: Always isolate the absolute value expression first before proceeding with solving the equation.
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Math Problem Analysis
Mathematical Concepts
Algebra
Absolute Value Equations
Extraneous Solutions
Formulas
|a| = b implies a = b or a = -b
Theorems
Absolute Value Property
Check for Extraneous Solutions
Suitable Grade Level
Grades 9-11