Solve for all values of yy in simplest form.
1, equals, open vertical bar, 8, minus, 2, y, close vertical bar
1=
∣8−2y∣

We are tasked with solving the equation:

\[
1 = |8 - 2y|
\]

To solve for \( y \), we must first recognize that an absolute value equation can have two cases, because absolute value makes both positive and negative expressions equal. This gives us two equations to solve:

### Case 1: \( 8 - 2y = 1 \)
\[
8 - 2y = 1
\]
Subtract 8 from both sides:
\[
-2y = 1 - 8
\]
\[
-2y = -7
\]
Now, divide both sides by -2:
\[
y = \frac{-7}{-2} = \frac{7}{2}
\]

### Case 2: \( 8 - 2y = -1 \)
\[
8 - 2y = -1
\]
Subtract 8 from both sides:
\[
-2y = -1 - 8
\]
\[
-2y = -9
\]
Now, divide both sides by -2:
\[
y = \frac{-9}{-2} = \frac{9}{2}
\]

### Solution:
The two solutions for \( y \) are:
\[
y = \frac{7}{2} \quad \text{or} \quad y = \frac{9}{2}
\]

Do you want any further details on this, or have any questions?

Here are some related questions:
1. How do you solve absolute value inequalities?
2. What happens when the absolute value is set equal to 0?
3. Can absolute value equations have no solutions?
4. How do you solve an absolute value equation involving fractions?
5. What are the geometric interpretations of absolute value equations?

**Tip:** Always remember to break absolute value equations into two separate cases, one for the positive and one for the negative.

Solve for all values of y in simplest form: 1 = |8 - 2y|.

Learn how to solve the absolute value equation 1 = |8 - 2y| with a clear, step-by-step solution. This problem explores the concept of absolute values and how to break them into two separate cases. Ideal for students in Grades 7-9.

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