Given the following equations: (i) $ x^2 - 4 = 0 $ (ii) $ y^2 - 9y + 20 = 0 $ Determine the correct relationship between $x$ and $y$. Choose the correct option: (a) $ x y $ (b) $ x \geq y $ (c) $ x

Question

Given the following equations:

(i) $ x^2 - 4 = 0 $

(ii) $ y^2 - 9y + 20 = 0 $

Determine the correct relationship between $x$ and $y$.

Choose the correct option:

(a) $ x > y $

(b) $ x \geq y $

(c) $ x < y $

(d) $ x \leq y $

(e) $ x = y $ or the relationship cannot be established

Accepted Answer

**Solution:**

1. Solve the first equation:
   $$
   x^2 - 4 = 0 \ 
   (x + 2)(x - 2) = 0
   $$
   So, $ x = 2 $ or $ x = -2 $.

2. Solve the second equation:
   $$
   y^2 - 9y + 20 = 0
   $$
   Factorizing:
   $$
   y^2 - 5y - 4y + 20 = 0 \ 
   y(y - 5) - 4(y - 5) = 0 \ 
   (y - 4)(y - 5) = 0
   $$
   So, $ y = 4 $ or $ y = 5 $.

3. Compare possible values of $x$ and $y$:
   - $x$ can be $2$ or $-2$
   - $y$ can be $4$ or $5$

4. Since both values of $y$ are greater than both values of $x$, the relationship is:
   $$
   x < y
   $$

**Answer:** (c) $ x < y $

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