Directions (41-45):In each of these questions, two equations (I) and (II) are given. Solve the equations and mark the correct option.

I. x² - 8x + 15 = 0
II. 2y² - 5y - 3 = 0

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

Step 1: Solve Equation I for $x$

$x^2-8x+15=0$

We look for two numbers whose product is $15$ and whose sum is $-8$. These numbers are $-3$ and $-5$.

$x^2-3x-5x+15=0$

$x(x-3)-5(x-3)=0$

$(x-3)(x-5)=0$

Therefore, the values of $x$ are:

$x=3,5$

Step 2: Solve Equation II for $y$

$2y^2-5y-3=0$

We look for two numbers whose product is $2\times (-3)=-6$ and whose sum is $-5$. These numbers are $-6$ and $1$.

$2y^2-6y+y-3=0$

$2y(y-3)+1(y-3)=0$

$(2y+1)(y-3)=0$

Therefore, the values of $y$ are:

$y=3,-\frac{1}{2}$

Step 3: Compare the values of $x$ and $y$

• Comparing $x=3$ with $y=3$: $x=y$

• Comparing $x=3$ with $y=-0.5$: $x>y$

• Comparing $x=5$ with $y=3$: $x>y$

• Comparing $x=5$ with $y=-0.5$: $x>y$

Combining all cases, we get $x\ge y$.

The correct option is If x ≥ y.