Find all values of $x$ which do not satisfy $$1 \le 4-3x+5x < 10-8x.$$ Express your answer in interval notation.

Answer 1

We have that $1 \le 4-3x+5x < 10-8x$ if and only if $1 \le 4+2x < 10-8x$. Next, we consider each of the two inequalities separately.

First, we solve $1 \le 4+2x$. Subtracting 4 from both sides gives $-3 \le 2x$, so dividing by 2 gives $-\frac{3}{2} \le x$.

Next, we solve $4+2x < 10-8x$. Adding $8x$ to both sides gives $8x+2x < 10-4$, or $10x<6$. Dividing by 10 gives $x<\frac{3}{5}$.

Therefore, $x$ satisfies both inequalities if and only if $-\frac35<x\le -\frac32$. Answer: $\boxed{\left(-\frac{3}{2}, -\frac{3}{5}\right)}$.