Question

-3 - |6 - 3x| + 7 = 4

Answer 1

New question. 4|7 - x| + 1 = -1

Answer 2

New question. |12 - 3x| = |9 + 4x| find x.

Answer 3

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

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

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

Now, we can isolate the absolute value:

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

\[
-|6 - 3x| = 0
\]

This implies:

\[
|6 - 3x| = 0
\]

The absolute value of a number is zero only when the number itself is zero. So, we can write:

\[
6 - 3x = 0
\]

Now, we can solve for \(x\):

\[
3x = 6
\]

\[
x = 2
\]

Now we can verify if this value of \(x\) satisfies the original equation:

Substituting \(x = 2\):

\[
-3 - |6 - 3(2)| + 7 = 4
\]

Calculating the absolute value:

\[
6 - 3(2) = 6 - 6 = 0
\]

So:

\[
-3 - |0| + 7 = -3 + 0 + 7 = 4
\]

The left side equals the right side, confirming that our solution is correct.

Thus, the solution is:

\[
\boxed{2}
\]

Answer 4

To solve the equation \(4|7 - x| + 1 = -1\), we will first isolate the absolute value term.

Subtract 1 from both sides of the equation:

\[
4|7 - x| = -1 - 1
\]

This simplifies to:

\[
4|7 - x| = -2
\]

Next, divide both sides by 4:

\[
|7 - x| = -\frac{2}{4}
\]

\[
|7 - x| = -\frac{1}{2}
\]

Since the absolute value of a real number cannot be negative, \( |7 - x| = -\frac{1}{2} \) has no solution.

Therefore, there are no values of \(x\) that satisfy the equation:

\[
\text{No solution.}
\]

Answer 5

To solve the equation \( |12 - 3x| = |9 + 4x| \), we will set up cases based on the properties of absolute values.

### Case 1: \(12 - 3x = 9 + 4x\)

1. Solve for \(x\):

\[
12 - 3x = 9 + 4x
\]

Rearranging gives:

\[
12 - 9 = 4x + 3x
\]

\[
3 = 7x
\]

Dividing both sides by 7:

\[
x = \frac{3}{7}
\]

### Case 2: \(12 - 3x = -(9 + 4x)\)

1. Solve for \(x\):

\[
12 - 3x = -9 - 4x
\]

Rearranging gives:

\[
12 + 9 = -4x + 3x
\]

\[
21 = -x
\]

Dividing by -1 gives:

\[
x = -21
\]

### Case 3: \(-(12 - 3x) = 9 + 4x\)

1. Solve for \(x\):

\[
-12 + 3x = 9 + 4x
\]

Rearranging gives:

\[
3x - 4x = 9 + 12
\]

\[
-x = 21
\]

Dividing by -1 gives:

\[
x = -21
\]

### Case 4: \(-(12 - 3x) = -(9 + 4x)\)

1. Solve for \(x\):

\[
-12 + 3x = -9 - 4x
\]

Rearranging gives:

\[
3x + 4x = -9 + 12
\]

\[
7x = 3
\]

Dividing by 7 gives:

\[
x = \frac{3}{7}
\]

### Summary of Solutions

From the four cases, we find two potential solutions:

- \( x = \frac{3}{7} \)
- \( x = -21 \)

Thus, the solutions to the equation \( |12 - 3x| = |9 + 4x| \) are:

\[
\boxed{\frac{3}{7}, -21}
\]