Least Common Multiple of 3 modulo equations

In general, you should use the Chinese Remainder Theorem.

However, for these particular numbers, there is a trick! Note that the congruences are equivalent to $2x+5\equiv 0\pmod{7}$, $2x+5\equiv 0\pmod{9}$, and $2x+5\equiv 0\pmod{11}$. So we want $2x+5$ to be divisible by $7$, $9$, and $11$, and therefore by $(7)(9)(11)$, which is $693$. So $2x+5=693$ will work. That gives $x=344$.

Remark: If you are not familiar with congruence notation, look at the sequences of numbers you produced. We will concentrate on the first one, which was $1,8,15,22,\dots$ (there was a mild typo).

Double these, and add $5$. We get $7,21, 35, 49,\dots$, the odd multiples of $7$. Similarly, when you double the numbers in your second sequence, and add $5$, you get the odd multiples of $9$. Similar treatment to the third sequence gets you the odd multiples of $11$. So we want $2x+5$ to be an odd multiple of $(7)(9)(11)$. The smallest positive one is $693$. So now solve $2x+5=693$.

You can also carry through the same argument using the language of remainders. We want $(2x+5)\%7=0$, $(2x+5)\%9=0$, and $(2x+5)\%11=0$.