I. ` x^(2)/2 + x - 1/2 = 1`
II. ` 3y^(2) - 10y + 8 = y^(2) + 2y - 10`

Options

A. ` x gt y`
B. ` x lt y`
C. ` x = y`
D. ` x ge y`

Explanation

Let's solve the equations step by step.

### Step 1: Solve the first equation for \( x \)

The first equation is:
\[ \frac{x^2}{2} + x - \frac{1}{2} = 1 \]

To eliminate the fraction, we can multiply the entire equation by 2:
\[ 2 \left(\frac{x^2}{2}\right) + 2x - 2\left(\frac{1}{2}\right) = 2 \cdot 1 \]
This simplifies to:
\[ x^2 + 2x - 1 = 2 \]

### Step 2: Rearrange the equation

Now, we will move all terms to one side:
\[ x^2 + 2x - 1 - 2 = 0 \]
This simplifies to:
\[ x^2 + 2x - 3 = 0 \]

### Step 3: Factor the quadratic equation

Next, we will factor the quadratic equation:
\[ x^2 + 3x - x - 3 = 0 \]
This can be factored as:
\[ (x + 3)(x - 1) = 0 \]

### Step 4: Find the values of \( x \)

Setting each factor to zero gives us:
\[ x + 3 = 0 \quad \Rightarrow \quad x = -3 \]
\[ x - 1 = 0 \quad \Rightarrow \quad x = 1 \]

Thus, the solutions for \( x \) are:
\[ x = -3 \quad \text{and} \quad x = 1 \]

### Step 5: Solve the second equation for \( y \)

The second equation is:
\[ 3y^2 - 10y + 8 = y^2 + 2y - 10 \]

First, we will move all terms to one side:
\[ 3y^2 - 10y + 8 - y^2 - 2y + 10 = 0 \]
This simplifies to:
\[ 2y^2 - 12y + 18 = 0 \]

### Step 6: Factor the quadratic equation

Now, we can factor out a common factor of 2:
\[ 2(y^2 - 6y + 9) = 0 \]
This can be factored as:
\[ 2(y - 3)(y - 3) = 0 \]

### Step 7: Find the values of \( y \)

Setting the factor to zero gives us:
\[ y - 3 = 0 \quad \Rightarrow \quad y = 3 \]

Thus, the solution for \( y \) is:
\[ y = 3 \quad \text{(with a multiplicity of 2)} \]

### Step 8: Determine the relationship between \( x \) and \( y \)

Now we have:
- \( x = -3 \) or \( x = 1 \)
- \( y = 3 \)

We need to find the relationship between \( x \) and \( y \):
1. For \( x = -3 \): \( y = 3 \) implies \( y > x \)
2. For \( x = 1 \): \( y = 3 \) also implies \( y > x \)

Thus, we conclude that:
\[ y > x \quad \text{or} \quad x < y \]

### Final Answer:
The relationship is \( x < y \).

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