$$-\frac{x+5}{2} \le \frac {12+3x}{4}$$
I always have issues with problems like this. I chose to ignore the negative sign at the beginning and got the answer. Is that a good method? When solving this inequality what is the best method for no mistakes?
My answer/ method: $$-(2)(12+3x) \le (4)(x+5)$$ $$-24-6x \le 4x+20$$ $$-24-20 \le 4x+6x$$ $$-44\le10x$$ $$-\frac{44}{10}\le x$$ $$-\frac{22}{5} \le x$$ $$\left(\infty,-\frac{22}{5}\right]$$
You have one mistake here may be typo -
$$-24+6x \le 4x+20$$
Its, $$-24-6x \le 4x+20$$
$$-24-20 \le 4x+6x$$
$$-44 \le 10x$$
$$\frac{-22}{5} \le x$$
As the value of $x \ge \frac{-22}{5}$
$$\left[\frac{-22}{5},\infty\right)$$
multiplying by $12$ we obtain $$-6(x+5)\le 3(12+3x)$$ multiplying out we get $$-6x-30\le 36+9x$$ thus we have $$-66\le 15x$$ can you proceed?
$$-\frac{x+5}{2} \le \frac {12+3x}{4}$$
Multiply $4$ both sides:
$$-2(x+5) \le 12+3x$$
$$-2x-10 \le 12 +3x$$
$$-22 \le 5x$$ $$x \geq \frac{-22}{5}$$
Your mistakes:
$-2(12+3x)=-24-6x$ rather than $-24+6x$. If this is just a typo, the next line is fine.
After this mistake, surprisingly in the next line, you corrected yourself.
$$x \ge -\frac{22}{5} \iff x \in [\frac{-22}{5},\infty)$$