Asked by 37432
Solve the absolute value equation -3 - |6 - 3x| + 7 = 4
x = -3
x = 0
x = 2
no solution
x = -3
x = 0
x = 2
no solution
Answers
Answered by
37432
Solve the absolute value equation -3 - |6 - 3x| + 7 = 4
x = -3
x = 0
x = 2
no solution
x = -3
x = 0
x = 2
no solution
Answered by
37432
5 |9 - 5n| -7 = 38
Answered by
37432
5 |9 - 5n| - 7 (not equal sign) 38
Answered by
37432
there is another answer
Answered by
37432
Solve the absolute value equation 4|7 - x| + 1 = -1
Answered by
37432
Solve the absolute value equation |12 - 3x| = |9 + 4x|
Answered by
37432
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 + ?|
y 2 1 0 1 2
Create an absolute value equation based on the table. (1 point)
y = |x + ?|
Answered by
37432
Solve the absolute value equation 2|x - 3| - 4 = -4
The solution to the absolute value equation is a x = ?
The solution to the absolute value equation is a x = ?
Answered by
37432
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).
The absolute value equation 7 |2x - 1| + 4 = 3 has ? solution(s).
Answered by
37432
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 = ?
The solutions to the absolute value equation are x = ? and x = ?
Answered by
37432
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.
Answered by
37432
its wrong
Answered by
GPT-4o mini
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:
\[
\boxed{2}
\]
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:
\[
\boxed{2}
\]
Answered by
GPT-4o mini
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:
\[
\boxed{2}
\]
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:
\[
\boxed{2}
\]
Answered by
GPT-4o mini
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:
\[
\boxed{0} \quad \text{and} \quad \boxed{3.6}
\]
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:
\[
\boxed{0} \quad \text{and} \quad \boxed{3.6}
\]
Answered by
GPT-4o mini
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 \(3.6\)).
### 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 \(3.6\)).
Answered by
GPT-4o mini
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:
\[
\boxed{(0, \frac{18}{5}) \cup (\frac{18}{5}, \infty)}
\]
### 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:
\[
\boxed{(0, \frac{18}{5}) \cup (\frac{18}{5}, \infty)}
\]
Answered by
GPT-4o mini
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:
\[
\boxed{\text{no solution}}
\]
### 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:
\[
\boxed{\text{no solution}}
\]
Answered by
GPT-4o mini
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:
\[
\boxed{\left\{\frac{3}{7}, -21\right\}}
\]
### 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:
\[
\boxed{\left\{\frac{3}{7}, -21\right\}}
\]
Answered by
GPT-4o mini
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.
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.
Answered by
GPT-4o mini
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:
\[
\boxed{x = 3}
\]
### 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:
\[
\boxed{x = 3}
\]
Answered by
GPT-4o mini
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:
\[
\boxed{0}
\]
### 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:
\[
\boxed{0}
\]
Answered by
GPT-4o mini
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:
\[
\boxed{1}
\]
### 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:
\[
\boxed{1}
\]
Answered by
GPT-4o mini
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.
\[
-|11x - 9| + 9 = 7
\]
### 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.
Answered by
GPT-4o mini
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!
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!
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