2^2x-1 = 32^x-2

2/x - 1/2x= 1/2
(2 over x minus 1 over 2x equals 1 over 2)

3/x + 8/x-5= 1
(3 over x plus 8 over x minus 5 equals 1)

2/x - 1/2x = 1/2

Make a common denominator on the left.
3/2x = 1/2
Cross-multiply.
2x = 6
Divide both sides by 2.
x = 3

Now you try one.

I am not sure what the exponents are in your first problem. If they are 2x-1 and x-2, rewrite the right side as a power of 2.
2^(2x-1) = (2^5)^(x-2) = 2^(5x-10)
The exponents of 2 must be equal.
2x - 1 = 5x - 10
3x = 9
x = 3

x over 4 is equal to negitive 1

To solve the equation 2^(2x-1) = 32^(x-2), we need to use the properties of exponents and logarithms.

Step 1: Express both sides of the equation with the same base. Since 2 and 32 are both powers of 2, we can rewrite 32 as 2^5:

2^(2x-1) = (2^5)^(x-2)

Step 2: Apply the power of a power rule, which states that (a^b)^c = a^(b*c).

2^(2x-1) = 2^(5(x-2))

Step 3: Since the bases are the same on both sides of the equation, we can equate the exponents:

2x - 1 = 5(x-2)

Step 4: Simplify both sides of the equation:

2x - 1 = 5x - 10

Step 5: Move all terms involving x to one side of the equation and the constants to the other side:

2x - 5x = -10 + 1

-3x = -9

Step 6: Divide both sides of the equation by -3 to solve for x:

x = -9 / -3

x = 3

Therefore, the solution to the equation 2^(2x-1) = 32^(x-2) is x = 3.

Now let's move on to the next equation.

To solve the equation 2/x - 1/2x = 1/2, we need to find a common denominator and combine the fractions.

Step 1: Find a common denominator for both fractions, which is 2x.

2(x/1) - (1/2)(1/1) = 1/2

Step 2: Convert both fractions to have the common denominator of 2x.

(2x(x/1) - (1/2)(x/1)) / 2x = 1/2

Step 3: Simplify the numerator:

(2x^2 - x/2) / 2x = 1/2

Step 4: Multiply both sides of the equation by 2x to eliminate the denominators:

2x^2 - x = 2(1/2)(2x)

2x^2 - x = 2x

Step 5: Move all terms to one side of the equation:

2x^2 - x - 2x = 0

2x^2 - 3x = 0

Step 6: Factor the quadratic equation or use the quadratic formula to solve for x.

x(2x - 3) = 0

x = 0 or 2x - 3 = 0

If x = 0, then the denominator 2/x becomes undefined, so it is not a valid solution.

Solving 2x - 3 = 0, we find:

2x = 3

x = 3/2

Therefore, the solution to the equation 2/x - 1/2x = 1/2 is x = 3/2.

Moving on to the last equation.

To solve the equation 3/x + 8/x - 5 = 1, we need to find a common denominator, combine the fractions, and solve for x.

Step 1: Find a common denominator for all three fractions, which is x.

(3/x) + (8/x) - 5 = 1

Step 2: Combine the fractions:

(3 + 8 - 5x) / x = 1

Step 3: Simplify the numerator:

(6 - 5x) / x = 1

Step 4: Multiply both sides of the equation by x to eliminate the denominator:

6 - 5x = (1)(x)

6 - 5x = x

Step 5: Move all terms involving x to one side of the equation:

6 = 6x

Step 6: Divide both sides of the equation by 6 to solve for x:

6/6 = x

x = 1

Therefore, the solution to the equation 3/x + 8/x - 5 = 1 is x = 1.