Directions (31-35): In each of the following questions two equations are given. Solve these equations and give answer:

I. x² + 12x + 35 = 0
II. y² + 9y + 20 = 0

The correct answer is if x ≤ y.

To determine the relationship between $x$ and $y$, we solve both quadratic equations step-by-step.

Step 1: Solve Equation I for $x$
The given equation is:
$x^2+12x+35=0$
We factorize the quadratic equation by splitting the middle term. We look for two numbers that multiply to 35 and add up to 12. These numbers are 7 and 5.
$x^2+7x+5x+35=0$
$x(x+7)+5(x+7)=0$
$(x+7)(x+5)=0$
This gives the roots:
$x=-7$ or $x=-5$

Step 2: Solve Equation II for $y$
The given equation is:
$y^2+9y+20=0$
We look for two numbers that multiply to 20 and add up to 9. These numbers are 5 and 4.
$y^2+5y+4y+20=0$
$y(y+5)+4(y+5)=0$
$(y+5)(y+4)=0$
This gives the roots:
$y=-5$ or $y=-4$

Step 3: Compare the values of $x$ and $y$
Let us compare each value of $x$ with each value of $y$:
- Comparing $x=-7$ with $y=-5$ and $y=-4$:
Since $-7<-5$ and $-7<-4$, we have $x<y$.
- Comparing $x=-5$ with $y=-5$ and $y=-4$:
Since $-5=-5$, we have $x=y$.
Since $-5<-4$, we have $x<y$.
Combining these comparisons, we find that $x$ is either less than or equal to $y$.
Thus, the relationship is:
$x\le y$