Solution
Inequality: Check Using Options
Given inequality:
\[ \frac{1}{x + 5} \leq \frac{1}{2x – 3} \]
Step 1: Try \( x = 0 \)
\[ \frac{1}{0 + 5} \leq \frac{1}{2(0) – 3} \Rightarrow \frac{1}{5} \leq \frac{1}{-3} \Rightarrow 0.2 \leq -0.333 \quad \text{(False)} \]
This means \( x = 0 \) does not satisfy the inequality.
– Option (1): \( -5 < x < \frac{3}{2} \) → includes 0 ✅
– Option (2): \( -5 < x < \frac{3}{2} \) → includes 0 ✅
– Option (3): does not include 0 ❌
– Option (4): does not include 0 ❌
So, eliminate Option 1 and Option 2.
Step 2: Try \( x = 100 \)
\[ \frac{1}{100 + 5} \leq \frac{1}{2(100) – 3} \Rightarrow \frac{1}{105} \leq \frac{1}{197} \Rightarrow 0.0095 \leq 0.0051 \quad \text{(False)} \]
This means \( x = 100 \) also does not satisfy the inequality.
– Option (3): includes \( x > \frac{3}{2} \), so includes 100 ✅
– Option (4): only includes up to \( x \leq 8 \) ❌
So, eliminate Option 3.
✅ Final Answer: Option (4)
\[ x < -5 \quad \text{or} \quad \frac{3}{2} < x \leq 8 \]
If \( m \) and \( n \) are natural numbers such that \( n > 1 \), and \( mn = 2^{25} \times 3^{40} \), then \( m – n \) equals:
Solution
✅ Explanation Summary
Given: \[ m^n = 2^{25} \cdot 3^{40} \]
This can be rewritten as: \[ m^n = (2^5 \cdot 3^8)^5 \] (since \( \gcd(25, 40) = 5 \), and \( n = 5 \))
Therefore: \[ m = 2^5 \cdot 3^8 = 32 \cdot 6561 = 209952 \quad \text{and} \quad n = 5 \]
So, the final answer is: \[ m – n = 209952 – 5 = \boxed{209947} \]
If x and y are real numbers such that 4x2 + 4y2 – 4xy – 6y + 3 = 0, then the value of (4x + 5y) is
Solution
✅ Direct Substitution Method (Faster)
We are given the equation:
\[ 4x^2 + 4y^2 – 4xy – 6y + 3 = 0 \]
We are asked to find the value of \( 4x + 5y \).
🔹 What is Direct Substitution?
Instead of solving algebraically, we substitute small values of \( x \) and \( y \) into the equation, and check:
- If the equation is satisfied
- Then compute \( 4x + 5y \) for those values
🔹 Try a few values:
- \( x = 0, y = 0 \Rightarrow 4(0)^2 + 4(0)^2 – 0 – 0 + 3 = 3 \quad \text{❌ Not 0}
- \( x = 1, y = 1 \Rightarrow 4 + 4 – 4 – 6 + 3 = 1 \quad \text{❌ Not 0}
- \( x = 0.5, y = 1 \Rightarrow 4(0.5)^2 + 4(1)^2 – 4(0.5)(1) – 6(1) + 3 = 1 + 4 – 2 – 6 + 3 = 0 \quad \text{✅ Works!}
🔹 Compute \( 4x + 5y \)
Using \( x = 0.5, y = 1 \): \[ 4x + 5y = 4(0.5) + 5(1) = 2 + 5 = \boxed{7} \]
✅ Final Answer: \( \boxed{7} \)
If x and y satisfy the equations |x| + x + y = 15 and x + |y| – y = 20, then (x – y) equals
1) 15 2) 10 3) 20 4) 5
Solution
Problem:
If \( x \) and \( y \) satisfy the equations: \[ |x| + x + y = 15 \quad \text{and} \quad x + |y| – y = 20, \] then find the value of \( x – y \).
Hit and Trial Solution:
Try \( x = 10 \), \( y = -5 \)
Check Equation 1:
\[
|x| + x + y = |10| + 10 + (-5) = 10 + 10 – 5 = 15 \quad \text{✓ Valid}
\]
Check Equation 2:
\[
x + |y| – y = 10 + 5 – (-5) = 10 + 5 + 5 = 20 \quad \text{✓ Valid}
\]
Since both equations are satisfied with \( x = 10 \) and \( y = -5 \), we compute: \[ x – y = 10 – (-5) = \boxed{15} \]
Final Answer:
\( \boxed{15} \)
Question
Solution
122
