Question

Saw the absolute value equation- 3 -| 6 - 3x| + 7 = 4

Answer 1

To solve the equation \( 3 - |6 - 3x| + 7 = 4 \), we first simplify the left side of the equation:

\[
3 + 7 - |6 - 3x| = 4
\]

This simplifies to:

\[
10 - |6 - 3x| = 4
\]

Next, we isolate the absolute value term by subtracting 10 from both sides:

\[
-|6 - 3x| = 4 - 10
\]

This simplifies to:

\[
-|6 - 3x| = -6
\]

Now, we multiply both sides by -1:

\[
|6 - 3x| = 6
\]

At this point, we can set up two cases based on the definition of absolute value:

1. \( 6 - 3x = 6 \)
2. \( 6 - 3x = -6 \)

**Case 1:**

\[
6 - 3x = 6
\]

Subtracting 6 from both sides gives:

\[
-3x = 0
\]

Dividing by -3:

\[
x = 0
\]

**Case 2:**

\[
6 - 3x = -6
\]

Subtracting 6 from both sides gives:

\[
-3x = -12
\]

Dividing by -3:

\[
x = 4
\]

Now we have two potential solutions:

\[
x = 0 \quad \text{and} \quad x = 4
\]

To verify, we can substitute both values back into the original equation:

1. For \( x = 0 \):

\[
3 - |6 - 3(0)| + 7 = 3 - |6| + 7 = 3 - 6 + 7 = 4
\]

This holds true.

2. For \( x = 4 \):

\[
3 - |6 - 3(4)| + 7 = 3 - |6 - 12| + 7 = 3 - |-6| + 7 = 3 - 6 + 7 = 4
\]

This also holds true.

Thus, the solutions to the equation are:

\[
\boxed{0} \quad \text{and} \quad \boxed{4}
\]