Solve the equation for all values of x.

open vertical bar, 4, x, minus, 4, close vertical bar, plus, 8, equals, 2, x

∣4*x*−4∣+8=

2*x*

The given equation is:

\[
|4x - 4| + 8 = 2x
\]

### Step 1: Isolate the absolute value expression

Subtract 8 from both sides of the equation:

\[
|4x - 4| = 2x - 8
\]

### Step 2: Solve for both cases of the absolute value

The absolute value \( |4x - 4| \) means that we need to solve two cases: one for when \( 4x - 4 \geq 0 \), and one for when \( 4x - 4 < 0 \).

#### Case 1: \( 4x - 4 \geq 0 \) or \( x \geq 1 \)

In this case, the absolute value is just \( 4x - 4 \), so the equation becomes:

\[
4x - 4 = 2x - 8
\]

Solve for \( x \):

\[
4x - 2x = -8 + 4
\]
\[
2x = -4
\]
\[
x = -2
\]

Since we assumed \( x \geq 1 \) in this case, the solution \( x = -2 \) does not satisfy this condition and should be discarded.

#### Case 2: \( 4x - 4 < 0 \) or \( x < 1 \)

In this case, the absolute value becomes \( -(4x - 4) = -4x + 4 \), so the equation becomes:

\[
-4x + 4 = 2x - 8
\]

Solve for \( x \):

\[
-4x - 2x = -8 - 4
\]
\[
-6x = -12
\]
\[
x = 2
\]

Since we assumed \( x < 1 \) in this case, the solution \( x = 2 \) does not satisfy this condition and should also be discarded.

### Conclusion:

There is no solution to the given equation because none of the values of \( x \) satisfy the original conditions.

Would you like further clarification or details on any step? Here are some related questions:

1. What is the meaning of absolute value in algebra?
2. How do you solve equations involving absolute values in general?
3. What happens when the solution doesn't fit the assumptions in a case analysis?
4. Can you explain why the solution set for this equation is empty?
5. How do we interpret the lack of solutions in the context of real numbers?

**Tip:** Always check your solutions against the original equation, especially when dealing with absolute values or inequalities.

Solve the equation for all values of x: |4x - 4| + 8 = 2x

Discover how to solve the absolute value equation |4x - 4| + 8 = 2x with a step-by-step solution. This problem covers essential algebraic concepts, including absolute value properties and linear equations, suitable for students in Grades 9-11. Understand why this equation has no valid solutions.

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