## Task 4 - 230934

Determine all real numbers $x$ for which the following holds:

$$
-5 \leq \frac{4 x-3}{2 x+1}<6
$$
It is $\frac{4 x-3}{2 x+1}=2-\frac{5}{2 x+1}$, so the inequality chain is equivalent to $-7-\frac{1}{2}$.

1. Case: $x>-\frac{1}{2}$: Then the first inequality is automatically satisfied and the second, after taking reciprocals, is equivalent to $2 x+1>\frac{5}{7}$, thus $x>-\frac{1}{7}$. Since $x>-\frac{1}{7}>-\frac{1}{2}$, all such $x$ (and no others that satisfy the case assumption) fulfill both inequalities.
2. Case: $x<-\frac{1}{2}$: Then the first inequality is automatically satisfied and the second, after taking reciprocals, is equivalent to $2 x+1<\frac{5}{7}$, thus $x<-\frac{9}{8}$. Since $x<-\frac{9}{8}<-\frac{1}{2}$, all such $x$ (and no others that satisfy the case assumption) fulfill both inequalities.

In summary, the inequality chain in the problem statement is satisfied exactly by those real numbers $x$ that are less than $-\frac{9}{8}$ or greater than $-\frac{1}{7}$.