We are solving the equation:

$|2x - 6| = |x|$

To solve this, we analyze the absolute values by breaking the equation into multiple cases.

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Step 1: Possible cases of absolute values

The definition of absolute values gives us two cases to solve. For $|2x - 6| = |x|$, the two possibilities are:

1. $2x - 6 = x$
2. $2x - 6 = -x$

We will solve both cases individually.

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Step 2: Solve Case 1: $2x - 6 = x$

1. Subtract $x$ from both sides: $2x - x - 6 = 0 \implies x - 6 = 0$
2. Solve for $x$: $x = 6$

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Step 3: Solve Case 2: $2x - 6 = -x$

1. Add $x$ to both sides: $2x + x - 6 = 0 \implies 3x - 6 = 0$
2. Solve for $x$: $3x = 6 \implies x = 2$

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Step 4: Verify solutions

We check whether both $x = 6$ and $x = 2$ satisfy the original equation $|2x - 6| = |x|$.

1. For $x = 6$: $|2(6) - 6| = |6| \implies |12 - 6| = |6| \implies |6| = |6| \quad \text{(True)}$

2. For $x = 2$: $|2(2) - 6| = |2| \implies |4 - 6| = |2| \implies |-2| = |2| \quad \text{(True)}$

Both solutions satisfy the equation.

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Final Answer:

The solutions are: $x = 6 \quad \text{and} \quad x = 2$

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Would you like more clarification on any of the steps? Let me know!

Related Questions:

1. How do you solve equations involving absolute values with variables?
2. What are the key properties of absolute value functions?
3. How do you graph absolute value functions like $|2x - 6|$ and $|x|$?
4. Can absolute value equations have no solution? Explain with examples.
5. How do you handle inequalities involving absolute values, like $|2x - 6| > |x|$?

Tip:

Always remember that absolute value splits an equation into two cases: one for the positive scenario and one for the negative scenario. Be sure to check all solutions in the original equation.