We need to expand the brackets on the right: -2x+3>6x-3. Add 2x to each side of the inequality: 3>8x-3 and add 3 to each side: 6>8x. Another way of writing this is 8x<6. Divide through by 2: 4x<3 and divide through by 4: x<3/4. We added 2x to each side because that helps us to bring the x's together. We added 3 to each side to bring the numbers together. Now we've ended up with x's on one side and a number on the other. Just what we want. 2 is common to 6 and 8 so we simplify the inequality by dividing through by it. Then to get x instead of 4x we divided through by 4. Why is 6>8x the same as 8x<6? Well, think about it. 2<3, isn't it? So 3>2. That's true. If Jack is taller than Jill, then Jill is shorter than Jack. See how it works? You just reverse the inequality when the quantities swap sides.
It's a good idea to check your answer by substituting a value for x in the original inequality, just to make sure we haven't made a mistake. The answer was that x must be less than 3/4 so let's try x=1/2. -2x+3 is -1+3=2. Now the right side: 2x-1 is 1-1=0 multiplied by 3 is still zero. Is the inequality correct? Yes it is, because 2 is greater than zero. If we put x=1 we should find that the inequality doesn't work out because x is bigger than 3/4. Let's see. The left is 3-2=1 and the right is 3. 1 isn't bigger than 3, so the inequality is false, as expected. If we put x=3/4 we will find that the left equals the right when it should be bigger than it. Looks like x<3/4 is right!