The behavior of the absolute value function changes at the two points where the absolute value expressions are equal to 0.
3x-4=0 --> x=4/3
2x+1=0 --> x=-1/2
Those two values of x divide the domain into three intervals: x less than -1/2, x between -1/2 and 4/3, and x greater than 4/3. Look for solutions in each of those intervals.
(1) x less than -1/2....
On this interval, and
That inequality is clearly never true; so there are no solutions in this interval.
(2) x between -1/2 and 4/3....
On this interval, and
x=-1/6 is in this interval, so part of the solution set is x between -1/6 and x=4/3
(3) x greater than 4/3....
On this interval, and
That inequality is always true, so the entire interval x greater than 4/3 is part of the solution set.
ANSWER: The solution set is (-1/6,infinity)
A graph confirms that answer:
The value of the absolute value function (red) is less than 7 (green) everywhere to the right of x=-1/6.
