2x-1 and x-1/2 are positive when x=>1/2 so the abs expressions behave as 2x-1 and x-1/2. Let's solve the inequality for x=>1/2: 2x-1+x-1/2-7<0. 3x-17/2<0, and 3x<17/2, so x<17/6. This puts x into the interval defined by 1/2<=x<17/6. This means x lies between 1/2 and 17/6 and can be 1/2 but must always be less than 17/6.
When x<1/2 the abs expressions behave like 1-2x and 1/2-x so the inequality becomes:
1-2x+1/2-x-7<0. -11/2-3x<0 so -11/2<3x and -11/6<x or x>-11/6. This puts x into the interval defined by -11/6<x<1/2. This means that x lies strictly between -11/6 and 1/2.
Note that, for example, x=0 is a valid solution because it lies between -11/6 and 1/2; x=1 is a valid solution because it lies between 1/2 and 17/6. These values both give 3/2<7, which is true.
The two intervals meet in the middle, so to speak, so are covered by the entire range or interval -11/6<x<17/6. The value x=1/2 gives -7<0 which is true.
What we learn from this is that absolute expressions put restrictions on solutions to equations or inequalities.