Math Problem Statement
∣x+6∣+6=3x
Solution
To solve the equation , follow these steps:
Step 1: Isolate the absolute value expression.
First, subtract 6 from both sides of the equation:
Step 2: Consider two cases for the absolute value.
The definition of absolute value gives two possible cases:
Case 1:
Solve for : Subtract from both sides: Add 6 to both sides: Divide by 2:
Case 2:
This simplifies to: Add to both sides: Add 6 to both sides: Divide by 4:
Step 3: Check for extraneous solutions.
Now, substitute the possible solutions and into the original equation to verify them.
-
For :
-
For :
Thus, is not a valid solution.
Final Answer:
The solution is .
Would you like more details or have any questions?
Here are some related questions:
- How do you handle absolute value equations with quadratic terms?
- What are some strategies for solving absolute value inequalities?
- Can there be extraneous solutions in other types of equations?
- How do we solve absolute value equations graphically?
- What happens if there's an absolute value on both sides of the equation?
Tip: Always check your solutions in the original equation, especially when dealing with absolute values, to avoid extraneous solutions!
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Math Problem Analysis
Mathematical Concepts
Algebra
Absolute Value Equations
Formulas
|a| = b implies a = b or a = -b
Theorems
Properties of Absolute Values
Suitable Grade Level
Grades 8-10