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

Answer 1

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 2

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

Answer 3

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 4

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

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:

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