Question

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+ ( )
|

Answer 1

Solve the absolute value equation 2|x−3|−4=−4
.(1 point)
The solution to the absolute value equation is x=( )

Answer 2

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 ( )solutions

Answer 3

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= ( )

Answer 4

To create an absolute value equation based on the given points in your table, we want to express \(y\) in terms of \(x\) using the absolute value function.

From the data provided:

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

We can observe that the values of \(y\) appear to create a "V" shape, typical for an absolute value function.

To determine the equation, we notice that the vertex of the "V" seems to occur when \(x = 1\), where \(y = 0\). This suggests that the equation involves the absolute value of \(x\) reflected around \(x = 1\).

We can express the equation as:

\[ y = |x - 1| \]

However, to shift it appropriately to match the given values:

We see that the peak appears to fit the form:

\[ y = 2 - |x - 1| \]

This equation fits the data:
- For \(x = -1\): \(y = 2 - |-1 - 1| = 2 - | -2 | = 2 - 2 = 0 \) → (Needs adjustment)

We will adjust our formula to:

**y = 2 - |x - 1|**

The correct value to fill in the parentheses is 1, thus the final adjustment can be made.

In conclusion, you can write the final equation as:

\[ y = 2 - |x - 1| \]

So it can also be stated as:

**Answer: y = |x + ( -1 )|**

Where the adjustment required for clarity would be the entire equation provided.

Answer 5

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

1. Add 4 to both sides of the equation:
\[
2|x - 3| - 4 + 4 = -4 + 4
\]
This simplifies to:
\[
2|x - 3| = 0
\]

2. Divide both sides by 2:
\[
|x - 3| = 0
\]

3. An absolute value equals zero only when the expression inside the absolute value is zero:
\[
x - 3 = 0
\]

4. Solving for \(x\):
\[
x = 3
\]

Thus, the solution to the absolute value equation is:
\[
x = 3
\]

### Answer:
x = (3)

Answer 6

To determine how many solutions exist for the absolute value equation \( 7|2x - 1| + 4 = 3 \), we will first isolate the absolute value term.

1. Subtract 4 from both sides:
\[
7|2x - 1| = 3 - 4
\]
Simplifying gives:
\[
7|2x - 1| = -1
\]

2. Divide both sides by 7:
\[
|2x - 1| = -\frac{1}{7}
\]

Since the absolute value of any expression is always non-negative, it cannot equal a negative number. Therefore, we conclude that there are **no solutions**.

### Final Answer:
The absolute value equation \( 7|2x - 1| + 4 = 3 \) has \( 0 \) solutions.

So the answer is:
0

Answer 7

To solve the absolute value equation \(-|11x - 9| + 9 = 7\), we start by isolating the absolute value expression.

1. Subtract 9 from both sides:
\[
-|11x - 9| + 9 - 9 = 7 - 9
\]
This simplifies to:
\[
-|11x - 9| = -2
\]

2. Multiply both sides by -1 to get:
\[
|11x - 9| = 2
\]

Now we will solve the two equations that arise from the absolute value:

### Case 1:
\[
11x - 9 = 2
\]
Adding 9 to both sides:
\[
11x = 11
\]
Dividing by 11:
\[
x = 1
\]

### Case 2:
\[
11x - 9 = -2
\]
Adding 9 to both sides:
\[
11x = 7
\]
Dividing by 11:
\[
x = \frac{7}{11}
\]

### Summary of Solutions
The solutions are \(x = 1\) and \(x = \frac{7}{11}\).

Since we need to provide the lower-valued solution first and the higher-valued solution second:

\[
x = \frac{7}{11} \quad \text{(lower solution)}
\]
\[
x = 1 \quad \text{(higher solution)}
\]

### Final Answer:
**The solutions to the absolute value equation are:**
- \( x = \frac{7}{11} \) (first response field)
- \( x = 1 \) (second response field)