Solve the Absolute Equation: Investigating |x + 4| = |2x + 20|

To solve the given absolute value equation $|x+4| = |2x+20|$, we consider the properties of absolute values:

• If $|A| = |B|$, then $A = B$ or $A = -B$.

Applying this, we consider two cases:

• Case 1: $x+4 = 2x+20$

• Case 2: $x+4 = - (2x+20)$

Let's solve each case:

Case 1:

$x+4 = 2x+20$

Rearrange the equation:
$x+4 = 2x+20 \implies x - 2x = 20 - 4$
$-x = 16$
So, $x = -16$.

Case 2:

$x+4 = - (2x+20)$

Distribute the negative sign:
$x+4 = -2x-20$

Rearrange the equation:
$x + 2x = -20 - 4$
$3x = -24$
$x = -8$.

Therefore, the solutions to the equation are $x = -8$ and $x = -16$.

To verify, plug back $x = -8$ and $x = -16$ into the original equation:

• For $x = -16$: $|x+4| = |-16 + 4| = 12$ and $|2x+20| = |2(-16) + 20| = 12$. Both sides equal.

• For $x = -8$: $|x+4| = |-8 + 4| = 4$ and $|2x+20| = |2(-8) + 20| = 4$. Both sides equal.

Both solutions satisfy the original equation. Therefore, the correct answer is $x = -16$ and $x = -8$.

Thus, the solution to the absolute value equation is:

The solutions are $\mathbf{x = -8}$ and $\mathbf{x = -16}$.