Solving Absolute Value Equations: Finding x in |x+3|=|2x+6|

To solve the equation $|x+3| = |2x+6|$, we need to consider the properties of absolute values and analyze the cases for different ranges of $x$.

The equation $|x+3| = |2x+6|$ implies two possibilities based on absolute value properties:

• $x+3 = 2x+6$
• $x+3 = -(2x+6)$

Let's solve each case:

Case 1: $x+3 = 2x+6$

Simplify the equation:
$x + 3 = 2x + 6$
Move $x$ to the other side:
$3 = x + 6$
Subtract 6 from both sides:
$x = -3$

Case 2: $x+3 = -(2x+6)$

Simplify the equation:
$x+3 = -2x-6$
Add $2x$ to both sides:
$3x + 3 = -6$
Subtract 3 from both sides:
$3x = -9$
Divide both sides by 3:
$x = -3$

In both cases, we find that $x = -3$. However, we need to verify if $x = -3$ satisfies the original equation:

Substitute $x = -3$ into the original equation:
$|x+3| = |2x+6|$
$|-3+3| = |2(-3)+6|$
$|0| = |-6+6|$
$0 = 0$

Therefore, $x = -3$ satisfies the equation. The solution to the problem is $x = -3$.

In conclusion, the correct answer is $x = -3$.