Solving Absolute Value Systems: |x + 3| = |2x + 6| with |x| = 3

To solve this system of equations, we will follow these steps:

• Step 1: Solve the second equation $|x| = 3$.
• Step 2: Check both potential solutions from Step 1 in the first equation $|x+3| = |2x+6|$.

Now, let's work through each step:
Step 1: Solve $|x| = 3$. This gives us two potential solutions: $x = 3$ and $x = -3$.

Step 2: Check these solutions in $|x+3| = |2x+6|$:
- For $x = 3$:
$|x+3| = |3+3| = |6| = 6$ and $|2x+6| = |2(3)+6| = |6+6| = |12| = 12$.
Since $6 \neq 12$, $x = 3$ does not satisfy the first equation.

- For $x = -3$:
$|x+3| = |-3+3| = |0| = 0$ and $|2x+6| = |2(-3)+6| = |-6+6| = |0| = 0$.
Both sides equal 0, so $x = -3$ satisfies the first equation.

Therefore, the solution to the system of equations is $x = -3$.