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Answer: 7

Using trial and improvement

Ones (units) column: E + E gives S

Tens column: E + E gives E

So E + E must be 10 or more, so that 1 is carried to give a different result in the two columns

Try E = 5

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   E can't be 5, because should have E in the tens column

Try E = 6

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   E should be larger

Try E = 8

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      E should be larger

Try E = 9

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That works, so E must be 9, and S must be 8. So we have 

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And X = 7.

Using algebra

Notice that the ones (units) column has $E + E$ and gives $S$, but in the tens column, $E + E$ gives $E$. So $E + E$ must be $10$ or more, so that $1$ is carried to give a different result in the tens column.

So $E + E = 10 + S$, and $E + E + 1 = 10 + E$.

$E + E + 1 = 10 + E$ means that $E + 1 = 10$, so $E = 9$.

So since $E + E = 10 + S$, $18 = 10 + S$, so $S = 8$.

So we have 

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And so $X=7$.