### Instruction: Find all $c$ which satisfy $$\frac{c}{3} \le 2+c < -2(1+c).$$Express your answer in interval notation, simplifying any fractions which occur in your answer.### Response: Let's first look at the left side of the inequality.
$\frac{c}{3} \le 2+c$ means $\frac{c}{3} - c \le 2$, or $\frac{-2c}{3} \le 2$.
That means $-2c \le 6$, or $c \ge -3$.
Now let's look at the right side of the inequality.
$2+c < -2(1+c)$ means $2+c < -2-2c$, or $3c < -4$.
That means $c < -4/3$.