Task 2 - 290822 a) Investigate whether the equation

$$
\left(\frac{x}{2}+2\right)(4 x-7)=2 x^{2}+\frac{x}{3}+1
$$

has a natural number $x$ as a solution!

b) In the given equation, the number 7 is to be replaced by a rational number $r$ so that the resulting equation has the number $x=1$ as a solution.

Determine all rational numbers $r$ that satisfy this requirement!
a) If $x$ is a solution to the given equation, then it follows that

$$
\begin{aligned}
2 x^{2}-\frac{7}{2} x+8 x-14 & =2 x^{2}+\frac{x}{3}+1 \\
48 x-21 x-2 x & =90 \\
5 x & =18
\end{aligned}
$$

Since this is not satisfied by any natural number $x$, the given equation has no natural number $x$ as a solution.

b) The resulting equation

$$
\left(\frac{x}{2}+2\right)(4 x-r)=2 x^{2}+\frac{x}{3}+1
$$

has $x=1$ as a solution if and only if $\left(\frac{1}{2}+2\right)(4-r)=2+\frac{1}{3}+1$ (1) holds. This is the case if and only if

$$
\frac{5}{2} \cdot(4-r)=\frac{10}{3} \quad \Rightarrow \quad r=\frac{8}{3}
$$

The given condition is thus satisfied precisely by the rational number $r=\frac{8}{3}$.