We have to take care of absolute values. The absolute value of an expression is the positive version of that expression, i.e.

So, first of all, we must observe that
is positive if

and similarly,

So, since
, we can divide the number line in three zones:
Zone 1: x < -8/5
In this zone, both expressions are negative. This means that

In fact, as we already said, the absolute value flips the sign of an expression if that expression is negative. So, the equation becomes

But we can't accept this solution (yet), because we're supposing x < -8/5.
Zone 2: -8/5 < x < -7/10
In this zone, 5x+8 has become positive, while 10x+7 is still negative. This means that

So, the equation becomes

Since indeed -8/5 < -1 < -7/10, we can accept this solution.
Zone 3: x > -7/10
In this zone, both expressions are positive, which means that the absolute value changes nothing:

So, the equation becomes

Since indeed 1/5 > -7/10, we can accept this solution.