x^2-12x+35>0; (x-7)(x-5)>0
Draw a number line and mark 7 and 5 on it.
The product of the factors x-7 and x-5 must be positive so x must be greater than 5 and 7. Therefore x must be further right (more positive than) 7 and 5, meaning that the segment of the number line immediately to the right of and beyond 7 satisfies the inequality x>7 and so x>5 because 7>5.
x-7 and x-5 can both be negative, so x<5 and x<7, so the whole segment on the left of 5 is also a solution and x<5. The segment between 5 and 7 (inclusive) does not satisfy the inequality.
To check the result, put x=5. This does not satisfy the inequality but x=4.9 does. Similarly x=7 does not satisfy the inequality but 7.1 does.