Solve the Absolute Value Equation: |x + 4| = |2x + 20| with x < -10

$\begin{cases} |x+4|=|2x+20| \\ x < -10 \end{cases}$

To solve the given problem, we'll work through these steps:

• Step 1: Solve the equation $|x+4| = |2x+20|$.
• Step 2: Consider the two cases derived from the absolute value equation.
• Step 3: Apply the condition $x < -10$ to determine the suitable solution.

Step 1: We start by considering the absolute value equation $|x+4| = |2x+20|$. This gives us two cases to explore:

• Case 1: $x+4 = 2x+20$
• Case 2: $x+4 = -(2x+20)$

Step 2: Solve each case individually:

Case 1:
Solving $x+4 = 2x+20$,
We first subtract $x$ from both sides to obtain:
$4 = x + 20$.
Subtracting 20 from both sides, we get:
$x = 4 - 20 \Rightarrow x = -16$.

Case 2:
Solving $x+4 = -(2x+20)$,
This simplifies to $x+4 = -2x - 20$.
Adding $2x$ to both sides, we have:
$3x + 4 = -20$.
Subtracting 4 from both sides gives:
$3x = -24$.
Dividing by 3, we find:
$x = -8$.

Step 3: Consider the inequality $x < -10$:

• Substitute $x = -16$ into the inequality: $-16 < -10$ is true.
• Substitute $x = -8$ into the inequality: $-8 < -10$ is false.

Therefore, the solution that satisfies both the equation and the inequality is $x = -16$.