If `5 ^(x) + (2 sqrt3) ^(2x) -169 le 0` is true for x lying in the interval :

Options

A. `(-oo,2)`
B. `(0,2)`
C. `(2,oo)`
D. `(0,4)`

Answer

To solve the inequality \( 5^x + (2\sqrt{3})^{2x} - 169 \leq 0 \), we can follow these steps:

### Step 1: Rewrite the expression
We start with the given inequality:
\[ 5^x + (2\sqrt{3})^{2x} - 169 \leq 0 \]
Notice that \( (2\sqrt{3})^{2x} = (2^2 \cdot 3)^{x} = 4^x \cdot 3^x \). Thus, we can rewrite the inequality as:
\[ 5^x + 4^x \cdot 3^x - 169 \leq 0 \]

### Step 2: Combine the terms
Next, we can express \( 4^x \cdot 3^x \) as \( (4 \cdot 3)^x = 12^x \):
\[ 5^x + 12^x - 169 \leq 0 \]

### Step 3: Rearrange the inequality
Now, we rearrange the inequality:
\[ 5^x + 12^x \leq 169 \]

### Step 4: Analyze the function
Let \( f(x) = 5^x + 12^x \). We need to find the values of \( x \) for which \( f(x) \leq 169 \).

### Step 5: Find critical points
To find critical points, we can evaluate \( f(x) \) at certain values:
- For \( x = 0 \):
\[ f(0) = 5^0 + 12^0 = 1 + 1 = 2 \]
- For \( x = 1 \):
\[ f(1) = 5^1 + 12^1 = 5 + 12 = 17 \]
- For \( x = 2 \):
\[ f(2) = 5^2 + 12^2 = 25 + 144 = 169 \]
- For \( x = 3 \):
\[ f(3) = 5^3 + 12^3 = 125 + 1728 = 1853 \]

### Step 6: Determine the interval
From the calculations:
- \( f(0) = 2 \) (less than 169)
- \( f(1) = 17 \) (less than 169)
- \( f(2) = 169 \) (equal to 169)
- \( f(3) = 1853 \) (greater than 169)

Since \( f(x) \) increases as \( x \) increases, we conclude that \( f(x) \leq 169 \) for \( x \) in the interval:
\[ (-\infty, 2] \]

### Final Answer
Thus, the solution to the inequality \( 5^x + (2\sqrt{3})^{2x} - 169 \leq 0 \) is:
\[ x \in (-\infty, 2] \]