Question

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

Answer 1

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

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

This simplifies to:

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

Now, isolate the absolute value term:

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

This leads to:

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

Next, we solve the absolute value equation \( |6 - 3x| = 6 \). This breaks down into two cases:

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

**Case 1:**

\[
6 - 3x = 6
\]
Subtracting 6 from both sides gives:

\[
-3x = 0 \implies x = 0
\]

**Case 2:**

\[
6 - 3x = -6
\]
Subtracting 6 from both sides gives:

\[
-3x = -12 \implies x = 4
\]

Thus, the solutions are \( x = 0 \) and \( x = 4 \).

Therefore, there are **2 solutions** to the equation \( 3 - |6 - 3x| + 7 = 4 \).