To find the value of x in the equation 3x-10 : 5x-10 = 1:2, we can follow a step-by-step approach:

1. Understanding the Equation:
The given equation is a proportion or ratio equation. It states that the ratio of 3x-10 to 5x-10 is equal to the ratio 1:2.

2. Cross-Multiplication:
Cross-multiplication is a technique used to solve proportion equations. We multiply the numerator of the first ratio with the denominator of the second ratio and vice versa. In this case, we have:
(3x-10) * 2 = (5x-10) * 1

3. Expanding and Simplifying:
Distribute the multiplication on both sides of the equation to eliminate the parentheses:
6x - 20 = 5x - 10

4. Isolating the Variable:
To solve for x, we need to isolate the variable on one side of the equation. We can do this by moving the constant terms to the other side of the equation.
Subtract 5x from both sides:
6x - 5x - 20 = -10

5. Simplifying Further:
Combine like terms on the left side of the equation:
x - 20 = -10

6. Isolating the Variable (Continued):
To isolate x, we need to get rid of the constant term on the left side. We can do this by adding 20 to both sides:
x - 20 + 20 = -10 + 20
x = 10

7. Checking the Solution:
It is always essential to verify the solution by substituting the found value of x back into the original equation.
using x = 10, we have:
3(10) - 10 : 5(10) - 10 = 1:2
30 - 10 : 50 - 10 = 1:2
20 : 40 = 1:2
Simplifying the ratio gives us 1:2, which confirms that x = 10 is the correct solution.

In conclusion, by following the step-by-step approach outlined above, we found that the value of x in the equation 3x-10 : 5x-10 = 1:2 is x = 10.

(3x-10) / (5x-10) = 1/2

cross multiply and solve

x= 10