Answer:
-4 < x < 1/2
Step-by-step explanation:
Given the inequality:
[tex]\displaystyle{\dfrac{x+4}{2x-1} < 0}[/tex]
Where x can not equal 1/2 since it will result the undefined expression. Furthermore, by solving a rational inequality (fractional inequality) is different from solving a linear inequality.
If we assume that for the expression on the left side is always positive, we can say that:
[tex]\displaystyle{\dfrac{x+4}{2x-1} < 0}[/tex]
However, the expression can remain negative. If a denominator remains somewhat negative and you multiply both sides by the denominator, you'll end up from < to >, basically swap in inequality sign.
Thus. If we let x > 1/2, the interval where the expression is always positive, we can solve the inequality as:
[tex]\displaystyle{\dfrac{x+4}{2x-1} \cdot \left(2x-1\right) < 0 \cdot \left(2x-1\right)}\\\\\displaystyle{x+4 < 0}\\\\\displaystyle{x < -4}[/tex]
However, this inequality is false so we can say that there's no region where this expression is less than 0 at x > 1/2.
Now let's say that at x < 1/2, the expression will start to remain in negative (although there is an interval that the expression is positive at x < 1/2 but majority negative.) Therefore, let's say:
[tex]\displaystyle{-\dfrac{x+4}{2x-1} < 0}\\\\\displaystyle{\dfrac{x+4}{2x-1} > 0}[/tex]
When we solve for the inequality, we should have x + 4 > 0 which results in x > -4. When union x > -4 with x < 1/2, it fits perfectly. Thus, the solution is:
[tex]\displaystyle{-4 < x < \dfrac{1}{2}}[/tex]
