Solve the Absolute Value Equation: Find x in |2x+4|+x-1=10

$|\left|2x+4\right|+x-1|=10$

To solve the equation $|\left|2x+4\right|+x-1|=10$, we will consider different cases due to the absolute value expressions.

First, let's set $|2x + 4| + x - 1 = 10$:

• Case 1: $2x + 4 \geq 0$ (so $|2x + 4| = 2x + 4$),
  Equation becomes: $2x + 4 + x - 1 = 10$,
  Combine and simplify: $3x + 3 = 10$,
  Solve: $3x = 7$ $\Rightarrow x = \frac{7}{3}$.
• Case 2: $2x + 4 < 0$ (so $|2x + 4| = -(2x + 4)$),
  Equation becomes: $-2x - 4 + x - 1 = 10$,
  Combine and simplify: $-x - 5 = 10$,
  Solve: $-x = 15$ $\Rightarrow x = -15$.

Next, consider $|2x + 4| + x - 1 = -10$:

• Case 3: $2x + 4 \geq 0$ (so $|2x + 4| = 2x + 4$),
  Equation becomes: $2x + 4 + x - 1 = -10$,
  Combine and simplify: $3x + 3 = -10$,
  Solve: $3x = -13$ $\Rightarrow x = -\frac{13}{3}$, which is not a solution since it doesn't satisfy the condition.
• Case 4: $2x + 4 < 0$ (so $|2x + 4| = -(2x + 4)$),
  Equation becomes: $-2x - 4 + x - 1 = -10$,
  Combine and simplify: $-x - 5 = -10$,
  Solve: $-x = -5$ $\Rightarrow x = 5$, which is not a solution since it doesn't satisfy the condition.

Thus, the possible solutions are $x = \frac{7}{3} \approx 2.3$ and $x = -15$.

Therefore, the solution to the problem is $x = 2.3$ and $x = -15$.