Find all values of $x$ such that $\frac{4x-5}{3x+5}\geq3$ Find all values of $x$ such that $\frac{4x-5}{3x+5}\geq3$.
If we consider two cases: $3x+5 > 0$ for the first and $3x + 5 < 0$ for the second.
When $3x+5 > 0$:
$\frac{4x-5}{3x+5}\geq3$.
Sloving for $x$ we find $x\leq -4$.
When $3x+5 < 0$:
$\frac{4x-5}{3x+5}\geq3$.
Sloving for $x$ we find $x\geq -4$
Additionaly through trail and error I discovered $x=-2$, $x=-3$ and $x=-4$... satisfy the original inequality.
$-2$, $-3$, $-4$ satisfy the inequality chain $-4 \leq x < -5/3$. All other values the satisify the inequality chain are also solutions.
In the case that $3x+5 > 0$, the inequality chain $-4 \geq x > -5/3$ is illogical.
Is this soultion correct? How can I improve it?
 A: $$\frac{4x-5}{3x+5}-3\ge0\\\implies\frac{4x-5-9x-15}{3x+5}\ge0\\\implies\frac{-5x-20}{3x+5}\ge0\\\implies\frac{x+4}{3x+5}\le0\\\implies x\in[-4,-\frac53)$$
EDIT: To understand the last step, let's take another example:
$$\frac{x+4}{x+2}\le0$$
We can't multiply the inequality with $(x+2)$ because this factor can be positive or negative. Accordingly, the sign of inequality will change. To prevent that, let's solve it differently.
If you put $x=-1,$ both  numerator and denominator remain positive, but we want the result to be negative. So, we figure $-1$ shouldn't be in our result set.
If we put $x=-5$, both numerator and denominator becomes negative, so overall the expression becomes positive but we want negative. So, we figure $-5$ shouldn't be in the result set either.
Now, what's special about $-1$ or $-5$? Both lie outside the interval $(-4,-2)$. End points of the interval are the points where our numerator and denominator are becoming zero.
If we put $x=-3$, numerator remains positive. Denominator becomes negative. So, overall the expression becomes negative. And this is what we want. And please note $-3$ lies inside the interval $(-4,-2)$.
So, our result set would have all the points from the inside of $(-4,-2)$ but none from the outside of it.
However, we don't just want our expression to be negative. It can be zero as well. And it would be zero at $-4$ only. Not at $-2$. (why?)
Thus, the final answer is $[-4,-2)$.
From this example, can you work out the solution to your question?
A: Your way to proceed also works, indeed we have

*

*for $3x+5>0 \iff x>-\frac 5 3$
$$\frac{4x-5}{3x+5}\geq3 \iff 4x-5\ge 9x+15 \iff 5x \le -20 \iff x \le -4$$
that is no solution for this first case, then

*

*for $3x+5<0 \iff x<-\frac 5 3$
$$\frac{4x-5}{3x+5}\geq3 \iff 4x-5\le 9x+15 \iff 5x \ge -20 \iff x \ge -4$$
that is $x \in \left[-4,-\frac 5 3\right)$.
