2x+1>9 and 5-3x<17.
Subtract 1 from each side of inequality 1: 2x>8, so x>4.
Add 3x to each side of inequality 2: 5<17+3x, and subtract 17 from each side: -12<3x, so, dividing through by 3, -4<x, and x>-4.
So we have x greater than both -4 and 4. But 4 is already greater than -4, so the solution is x>4, because it satisfies both inequalities.
However, if the question requires only one inequality of the two to be true, then x>4 or x>-4. Therefore it's important that the question establishes whether “and” or “or” is actually meant. “Or” means there's no absolute solution, so, for example, x=0 would satisfy one inequality but not the other and x=5 satisfies both.