Solve for all values of xx in simplest form.
open vertical bar, minus, 4, x, plus, 4, close vertical bar, equals, 12
∣−4x+4∣=
12

To solve the equation \(\left|-4x + 4\right| = 12\), we need to consider the definition of absolute value. The absolute value of a number \(a\) is defined as:

\[
\left| a \right| = 
\begin{cases} 
a & \text{if } a \geq 0 \\
-a & \text{if } a < 0 
\end{cases}
\]

Given the equation \(\left|-4x + 4\right| = 12\), this means:

1. \(-4x + 4 = 12\)
2. \(-4x + 4 = -12\)

We will solve each case separately.

### Case 1: \(-4x + 4 = 12\)

\[
-4x + 4 = 12
\]

Subtract 4 from both sides:

\[
-4x = 8
\]

Now, divide both sides by \(-4\):

\[
x = -2
\]

### Case 2: \(-4x + 4 = -12\)

\[
-4x + 4 = -12
\]

Subtract 4 from both sides:

\[
-4x = -16
\]

Now, divide both sides by \(-4\):

\[
x = 4
\]

### Final Solution

The solutions to the equation are \(x = -2\) and \(x = 4\).

Would you like any further details or have any questions?

Here are 5 related questions:
1. How would the solution change if the absolute value equation was \(\left|2x - 3\right| = 7\)?
2. What happens if the equation had no real solutions? How would that be determined?
3. How can absolute value inequalities be solved similarly to absolute value equations?
4. Can you solve an absolute value equation with variables on both sides, like \(\left|2x - 3\right| = \left|x + 4\right|\)?
5. How would you graph the solution to \(\left|-4x + 4\right| = 12\)?

**Tip:** When solving absolute value equations, always consider both the positive and negative cases to ensure all possible solutions are found.

Learn how to solve the absolute value equation |−4x + 4| = 12 with a step-by-step solution. Understand the definition of absolute value and how to solve equations involving absolute value. Suitable for Grades 9-12 students.

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