Solve:
a) 1/(2^x) = 1/(x+2)
b) 1/(2^x) > 1/(x^2)
To solve these equations, we need to isolate the variable x. Let's start with solving equation (a).
a) 1/(2^x) = 1/(x+2)
First, we can eliminate the denominators by cross-multiplying:
(x+2) = 2^x
To continue, we can rewrite 2^x as 2 raised to the power of x. Let's find a common base to equate both sides of the equation:
2^x = 2^(x+2)
Since the bases are the same, the exponents must be equal:
x = x + 2
Now, subtracting x from both sides, we get:
0 = 2
We have reached an inconsistent statement, 0 = 2, which means there are no solutions for this equation. Hence, equation (a) has no solution.
Moving on to equation (b).
b) 1/(2^x) > 1/(x^2)
First, let's eliminate the denominators by cross-multiplying:
x^2 > 2^x
We can rewrite 2^x as 2 raised to the power of x.
x^2 > 2^(x)
Next, let's consider different cases to solve the inequality.
Case 1: When x > 0
For positive values of x, we know that the function 2^x is increasing in value. Since x^2 is always positive, the inequality x^2 > 2^x holds true for all positive values of x. Therefore, the set of solutions for this case is x > 0.
Case 2: When x < 0
For negative values of x, we know that the function 2^x is decreasing in value. However, 2^x cannot be negative for any real number x. Hence, x^2 > 2^x is false for all negative values of x.
Case 3: When x = 0
By substituting x = 0 into the inequality, we get:
0^2 > 2^0
0 > 1
This is false, so x = 0 is not a solution.
Therefore, the solution to the inequality x^2 > 2^x is x > 0.
In summary:
a) Equation (a) has no solution.
b) The solution to equation (b) is x > 0.