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

To solve the absolute value equation \(-3 - |6 - 3x| + 7 = 4\), we first simplify the equation.

Starting from the original equation:
\[
-3 - |6 - 3x| + 7 = 4
\]

We combine the constants \(-3\) and \(7\):
\[
4 - |6 - 3x| = 4
\]

Next, we isolate the absolute value term:
\[
-|6 - 3x| = 0
\]

This implies:
\[
|6 - 3x| = 0
\]

An absolute value is equal to zero when its argument is zero:
\[
6 - 3x = 0
\]

Solving for \(x\):
\[
3x = 6 \implies x = 2
\]

Next, we check if \(x = 2\) is a solution by substituting it back into the original equation:
\[
-3 - |6 - 3(2)| + 7 = 4
\]
Calculating the absolute value:
\[
6 - 6 = 0 \implies |6 - 3 \cdot 2| = 0
\]
Thus, we have:
\[
-3 - 0 + 7 = 4
\]
\[
4 = 4
\]

Since this is true, the solution is confirmed to be correct.

Thus, the solution is:
\[
\boxed{2}
\]

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

x = -3
x = 0
x = 2
no solution

To solve the equation \(-3 - |6 - 3x| + 7 = 4\), we start by simplifying the left-hand side.

1. Combine the constants \(-3\) and \(+7\):
\[
4 - |6 - 3x| = 4
\]

2. Since both sides equal \(4\), we can simplify the equation to:
\[
-|6 - 3x| = 0
\]

3. This implies:
\[
|6 - 3x| = 0
\]

4. The absolute value is zero when its argument is zero, so we set up the equation:
\[
6 - 3x = 0
\]

5. Solving for \(x\):
\[
3x = 6 \implies x = 2
\]

Now, we will check if \(x = 2\) satisfies the original equation:

6. Substitute \(x = 2\) back into the original equation:
\[
-3 - |6 - 3(2)| + 7 = 4
\]
Simplifying inside the absolute value:
\[
6 - 6 = 0 \implies |6 - 6| = 0
\]
Substitute back into the equation:
\[
-3 - 0 + 7 = 4
\]
This simplifies to:
\[
4 = 4
\]

Since this is true, \(x = 2\) is indeed a solution.

Thus, the final answer is:
\[
\boxed{2}
\]

5 |9 - 5n| -7 = 38

To solve the equation \(5 |9 - 5n| - 7 = 38\), follow these steps:

1. **Isolate the absolute value**:
Add \(7\) to both sides:
\[
5 |9 - 5n| = 45
\]

2. **Divide by \(5\)**:
\[
|9 - 5n| = 9
\]

3. **Set up two cases** for the absolute value:
- Case 1: \(9 - 5n = 9\)
- Case 2: \(9 - 5n = -9\)

4. **Solve Case 1**:
\[
9 - 5n = 9
\]
Subtract \(9\) from both sides:
\[
-5n = 0 \implies n = 0
\]

5. **Solve Case 2**:
\[
9 - 5n = -9
\]
Subtract \(9\) from both sides:
\[
-5n = -18
\]
Divide by \(-5\):
\[
n = \frac{-18}{-5} = \frac{18}{5} = 3.6
\]

6. **Summary of solutions**:
The solutions to the equation are:
\[
n = 0 \quad \text{and} \quad n = 3.6
\]

So, the final answers are:
\[
\boxed{0} \quad \text{and} \quad \boxed{3.6}
\]

5 |9 - 5n| - 7 (not equal sign) 38

It looks like you want to solve the inequality \(5 |9 - 5n| - 7 < 38\). Let's go through the steps to solve this inequality.

### Step 1: Isolate the absolute value

Start by adding \(7\) to both sides of the inequality:
\[
5 |9 - 5n| < 38 + 7
\]
\[
5 |9 - 5n| < 45
\]

Now, divide both sides by \(5\):
\[
|9 - 5n| < 9
\]

### Step 2: Set up the corresponding inequalities

The absolute value inequality \(|A| < B\) translates to:
\[
-B < A < B
\]
In our case:
\[-9 < 9 - 5n < 9\]

### Step 3: Solve the compound inequality

1. **First part**:
\[
-9 < 9 - 5n
\]
Subtract \(9\) from both sides:
\[
-18 < -5n
\]
Divide by \(-5\) (remember to flip the inequality sign):
\[
\frac{18}{5} > n \quad \text{(or)} \quad n < \frac{18}{5}
\]

2. **Second part**:
\[
9 - 5n < 9
\]
Subtract \(9\) from both sides:
\[
-5n < 0
\]
Divide by \(-5\) (again flipping the inequality sign):
\[
n > 0
\]

### Step 4: Combine the results

From the inequalities we have:
\[
0 < n < \frac{18}{5}
\]

### Final Answer

The solution to the inequality \(5 |9 - 5n| - 7 < 38\) is:
\[
\boxed{(0, \frac{18}{5})}
\]

This means \(n\) can take any value between \(0\) and \(3.6\) (not including \(0\) and \(3.6\)).

there is another answer

Let’s reevaluate the inequality \(5 |9 - 5n| - 7 < 38\).

### Step 1: Isolate the absolute value

Add \(7\) to both sides:
\[
5 |9 - 5n| < 45
\]
Now, divide both sides by \(5\):
\[
|9 - 5n| < 9
\]

### Step 2: Set up the inequality

An absolute value inequality of the form \(|A| < B\) translates to:
\[
-B < A < B
\]
For our case:
\[
-9 < 9 - 5n < 9
\]

### Step 3: Solve the two parts of the compound inequality

#### Part 1:
\[
-9 < 9 - 5n
\]
Subtract \(9\) from both sides:
\[
-18 < -5n
\]
Now, divide by \(-5\) (remember to flip the inequality):
\[
\frac{18}{5} > n \quad \text{(or equivalently)} \quad n < \frac{18}{5}
\]

#### Part 2:
\[
9 - 5n < 9
\]
Subtract \(9\) from both sides:
\[
-5n < 0
\]
Now divide by \(-5\) (again flipping the inequality):
\[
n > 0
\]

### Step 4: Combine the results

So far, we have:
- From Part 1: \(n < \frac{18}{5} \) (or \(n < 3.6\))
- From Part 2: \(n > 0\)

### Final Interval Solution
Combining these inequalities gives us:
\[
0 < n < \frac{18}{5}
\]

So the solution is:
\[
(n \in (0, 3.6))
\]

### Finding the Other Case
We need to take into account the case for the absolute value when it is negative.

### Step 5: Absolute Value Greater Than

In addition to the less than case, we also look at when \( |9 - 5n| \) is greater than \(9\):
\[
|9 - 5n| > 9
\]

This splits into two cases:

1. **Case 1**:
\[
9 - 5n > 9
\]
\(-5n > 0\) thus \(n < 0\) (This will give us negative values)

2. **Case 2**:
\[
9 - 5n < -9
\]
Rearranging gives us:
\[
-5n < -18 \implies n > \frac{18}{5}
\]

### Final Combined Solutions
Combining both parts:

1. From the case \(0 < n < \frac{18}{5}\)
2. From the case \(n < 0\) (which is out of the valid \(n\) range)
3. And \(n > \frac{18}{5}\), which means \(n > 3.6\)

So, the complete solution set is:
- \( (0, \frac{18}{5}) \)
- \( (3.6, \infty) \)

Thus, the complete solution can be represented as:
\[
\boxed{(0, \frac{18}{5}) \cup (\frac{18}{5}, \infty)}
\]

Solve the absolute value equation 4|7 - x| + 1 = -1

To solve the equation \(4|7 - x| + 1 = -1\), follow these steps:

### Step 1: Isolate the absolute value

First, we can subtract \(1\) from both sides of the equation:
\[
4|7 - x| = -1 - 1
\]
\[
4|7 - x| = -2
\]

### Step 2: Consider the properties of absolute values

Since the absolute value \(|A|\) is always non-negative (i.e., \(|A| \geq 0\) for any \(A\)), the left-hand side, which is \(4|7 - x|\), must also be non-negative. However, we set it equal to \(-2\), which is impossible.

### Conclusion

Since we cannot have \(4|7 - x| = -2\), the equation has **no solutions**.

Thus, the final answer is:
\[
\boxed{\text{no solution}}
\]

Solve the absolute value equation |12 - 3x| = |9 + 4x|

To solve the equation \(|12 - 3x| = |9 + 4x|\), we need to break it down into cases based on the definitions of absolute values.

### Two Cases to Consider:

1. \(12 - 3x = 9 + 4x\)
2. \(12 - 3x = -(9 + 4x)\)

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

1. Rearranging gives:
\[
12 - 9 = 4x + 3x
\]
\[
3 = 7x
\]
\[
x = \frac{3}{7}
\]

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

1. Distributing the negative sign gives:
\[
12 - 3x = -9 - 4x
\]

2. Rearranging:
\[
12 + 9 = -4x + 3x
\]
\[
21 = -x
\]
\[
x = -21
\]

### Summary of Solutions

We have two potential solutions from the two cases:
1. \(x = \frac{3}{7}\)
2. \(x = -21\)

### Verify Solutions

**For \(x = \frac{3}{7}\)**:
\[
12 - 3\left(\frac{3}{7}\right) = 12 - \frac{9}{7} = \frac{84}{7} - \frac{9}{7} = \frac{75}{7}
\]
\[
9 + 4\left(\frac{3}{7}\right) = 9 + \frac{12}{7} = \frac{63}{7} + \frac{12}{7} = \frac{75}{7}
\]
Thus, \(|12 - 3x| = |9 + 4x|\) holds true.

**For \(x = -21\)**:
\[
12 - 3(-21) = 12 + 63 = 75
\]
\[
9 + 4(-21) = 9 - 84 = -75
\]
Thus, \(|12 - 3x| = |9 + 4x|\) holds true as well.

### Final Answer
Both solutions are valid:
\[
\boxed{\left\{\frac{3}{7}, -21\right\}}
\]

X -1 0 1 2 3

y 2 1 0 1 2

Create an absolute value equation based on the table. (1 point)
y = |x + ?|

To create an absolute value equation based on the given table of values, we are looking for a form of the equation \(y = |x + b|\) that matches the values of \(y\) for the corresponding values of \(x\).

Let's start by analyzing the points provided:

- For \(x = -1\), \(y = 2\)
- For \(x = 0\), \(y = 1\)
- For \(x = 1\), \(y = 0\)
- For \(x = 2\), \(y = 1\)
- For \(x = 3\), \(y = 2\)

### Step 1: Find vertex of the absolute value function

From the table, we can see that the absolute value graph has its vertex at \(x = 1\) where \(y = 0\). This means the expression inside the absolute value should be \(x - 1\) (shifting the graph to the right).

### Step 2: Formulate the equation

We can start with:
\[
y = |x - 1|
\]

### Step 3: Adjust for the vertical shift

However, we can see that for \(x = 0\), \(y\) is \(1\), which indicates that we may need to adjust our equation vertically.

To shift down, we can write:
\[
y = |x - 1| + 0
\]

This to start matches the maximum point around \(y = 0\), but we see that the values of \(y\) can also be adjusted, as in fact, the lowest point is exactly at \(y = 0\).

### Final Absolute Value Equation

Thus, the final equation that fits the given points is:
\[
\boxed{y = |x - 1|}
\]

To match your format \(y = |x + b|\), we can rewrite \(x - 1\) as:
\[
y = |x + (-1)| \implies b = -1
\]
Hence:
\[
\boxed{y = |x - 1| \text{ or } y = |x + (-1)|}
\]
This confirms the correct points as per your provided data table.

Solve the absolute value equation 2|x - 3| - 4 = -4

The solution to the absolute value equation is a x = ?

To solve the absolute value equation \(2|x - 3| - 4 = -4\), start by isolating the absolute value term.

### Step 1: Isolate the absolute value

Add \(4\) to both sides of the equation:
\[
2|x - 3| - 4 + 4 = -4 + 4
\]
\[
2|x - 3| = 0
\]

### Step 2: Divide by \(2\)

Now, divide both sides by \(2\):
\[
|x - 3| = 0
\]

### Step 3: Solve the absolute value equation

The absolute value is equal to \(0\) only when the expression inside is equal to \(0\):
\[
x - 3 = 0
\]

Adding \(3\) to both sides gives:
\[
x = 3
\]

### Conclusion

The solution to the absolute value equation is:
\[
\boxed{x = 3}
\]

Solve the absolute value equation 7 |2x - 1| + 4 = 3 to determine how many solutions exist. Enter 1 for one solution, 2 for two solutions, or 0 for no solutions. (1 point)

The absolute value equation 7 |2x - 1| + 4 = 3 has ? solution(s).

To solve the absolute value equation \(7 |2x - 1| + 4 = 3\), we first isolate the absolute value term.

### Step 1: Isolate the absolute value

Subtract \(4\) from both sides:
\[
7 |2x - 1| = 3 - 4
\]
\[
7 |2x - 1| = -1
\]

### Step 2: Analyze the equation

Since the absolute value \(|2x - 1|\) cannot be negative, multiplying it by \(7\) still results in a non-negative number. This means that:
\[
7 |2x - 1| \geq 0
\]

Thus, \(7 |2x - 1| = -1\) has no possible solutions because the left-hand side is always zero or positive, while the right-hand side is negative.

### Conclusion

Therefore, the absolute value equation \(7 |2x - 1| + 4 = 3\) has **0 solutions**.

Thus, the answer is:
\[
\boxed{0}
\]

Solve the absolute value equation - |11x - 9| + 9 = 7. Enter the lower-valued solution in the first response field and the higher-valued solution in the second response field. Provide exact solutions in fractional form, where appropriate. (1 point)

The solutions to the absolute value equation are x = ? and x = ?

To solve the absolute value equation \(-|11x - 9| + 9 = 7\), we will follow these steps:

### Step 1: Isolate the absolute value

Subtract \(9\) from both sides:
\[
-|11x - 9| + 9 - 9 = 7 - 9
\]
\[
-|11x - 9| = -2
\]

### Step 2: Multiply by \(-1\)

Multiplying both sides by \(-1\) (which flips the inequality):
\[
|11x - 9| = 2
\]

### Step 3: Set up the cases for the absolute value

The equation \(|11x - 9| = 2\) gives us two cases to consider:

**Case 1:**
\[
11x - 9 = 2
\]

**Case 2:**
\[
11x - 9 = -2
\]

### Step 4: Solve each case

**Solving Case 1:**
\[
11x - 9 = 2
\]
Add \(9\) to both sides:
\[
11x = 11
\]
Divide by \(11\):
\[
x = 1
\]

**Solving Case 2:**
\[
11x - 9 = -2
\]
Add \(9\) to both sides:
\[
11x = 7
\]
Divide by \(11\):
\[
x = \frac{7}{11}
\]

### Step 5: Summary of solutions

We have two solutions:

1. \(x = 1\)
2. \(x = \frac{7}{11}\)

### Final Ordering of Solutions

The lower-valued solution is \(\frac{7}{11}\) and the higher-valued solution is \(1\).

Thus, the answers are:
- Lower-valued solution:
\[
\boxed{\frac{7}{11}}
\]
- Higher-valued solution:
\[
\boxed{1}
\]

To identify the solutions to the absolute value equation, start by Isolating the absolute value expression by applying inverse operations. Once isolated, split the absolute value expression into two equations and solve. Make sure to provide exact solutions in fractional form and to enter the lower-valued solution in the first response field.

Let's solve the absolute value equation given the general method you've outlined. The specific equation we'll consider is:

\[
-|11x - 9| + 9 = 7
\]

### Step 1: Isolate the absolute value expression

First, we will isolate the absolute value. Start by subtracting \(9\) from both sides:
\[
-|11x - 9| + 9 - 9 = 7 - 9
\]
\[
-|11x - 9| = -2
\]

Next, multiply both sides by \(-1\) (which will also flip the sign):
\[
|11x - 9| = 2
\]

### Step 2: Split the absolute value expression into two equations

The equation \(|11x - 9| = 2\) gives us two cases to consider.

#### Case 1:
\[
11x - 9 = 2
\]

Add \(9\) to both sides:
\[
11x = 11
\]

Now, divide by \(11\):
\[
x = 1
\]

#### Case 2:
\[
11x - 9 = -2
\]

Add \(9\) to both sides:
\[
11x = 7
\]

Now, divide by \(11\):
\[
x = \frac{7}{11}
\]

### Step 3: Summary of solutions

We have found two solutions:

1. \(x = 1\)
2. \(x = \frac{7}{11}\)

### Step 4: Identify the lower-valued and higher-valued solutions

The lower-valued solution is \(\frac{7}{11}\) and the higher-valued solution is \(1\).

#### Final Answers:
- Lower-valued solution:
\[
\boxed{\frac{7}{11}}
\]
- Higher-valued solution:
\[
\boxed{1}
\]

This method clearly isolates the absolute value, splits it into equations, and finds exact solutions in fractional form as requested.

its wrong

Let’s carefully go through the problem step by step to ensure we find the correct solutions for the absolute value equation:

Given:

\[
-|11x - 9| + 9 = 7
\]

### Step 1: Isolate the absolute value expression

Subtract \(9\) from both sides:

\[
-|11x - 9| = 7 - 9
\]
\[
-|11x - 9| = -2
\]

Now multiply both sides by \(-1\):

\[
|11x - 9| = 2
\]

### Step 2: Split into two cases

Now we have two cases based on the absolute value definition:

#### Case 1:

\[
11x - 9 = 2
\]

Add \(9\) to both sides:

\[
11x = 11
\]

Now, divide by \(11\):

\[
x = 1
\]

#### Case 2:

\[
11x - 9 = -2
\]

Add \(9\) to both sides:

\[
11x = 7
\]

Now, divide by \(11\):

\[
x = \frac{7}{11}
\]

### Step 3: Summary of the solutions

From the two cases, we found:

1. \(x = 1\)
2. \(x = \frac{7}{11}\)

### Step 4: Identify the lower-valued and higher-valued solutions

- The lower-valued solution is \(\frac{7}{11}\).
- The higher-valued solution is \(1\).

### Final Answers

- Lower-valued solution:
\[
\boxed{\frac{7}{11}}
\]

- Higher-valued solution:
\[
\boxed{1}
\]

This confirms the steps and aligns with the process of isolating the absolute value, splitting into cases, and solving the resulting equations. If you have a different original equation or specific conditions, please let me know, and I'll help accordingly!