Disequazioni letterali intere es 3-4

[stefano166]

(x+2a)(x-2a) >=1-4a

x(x+4)-k(k+6)-5 > 0

[ce8c0e31f941cc14ed8def9a9c5c3a60a8400d85]

3.

[math]
\begin{aligned}
& (x + 2\,a)(x - 2\,a) \ge 1 - 4\,a \\
& x^2 - 4\,a^2 + 4\,a - 1 \ge 0 \\
& x^2 - (2\,a - 1)^2 \ge 0 \\
& (x + 2\,a - 1)(x - 2\,a + 1) \ge 0 \\
& \text{se} \; a < \frac{1}{2}: x \le 2\,a - 1 \, \vee \, x \ge 1 - 2\,a \; ; \\
& \text{se} \; a = \frac{1}{2} : x = 0 \; ; \\
& \text{se} \; a > \frac{1}{2} : x \le 1 - 2\,a \, \vee \, x \ge 2\,a - 1 \; .
\end{aligned} \\
[/math]

4.

[math]
\begin{aligned}
& x\,(x + 4) - k\,(k + 6) - 5 > 0 \\
& x^2 + 4\,x - k^2 - 6\,k - 5 > 0 \\
& x^2 + (k + 5)\,x - (k + 1)\,x - (k + 5)(k + 1) > 0 \\
& x\,(x + k + 5) - (k + 1)\,(x + k + 5) > 0 \\
& (x - k - 1)(x + k + 5) > 0 \\
& \text{se} \; k < - 3 : x < k + 1 \, \vee \, x > - k - 5 \; ; \\
& \text{se} \; k = - 3 : x \ne - 2 \; ; \\
& \text{se} \; k > - 3 : x < - k - 5 \, \vee \, x > k + 1 \; .
\end{aligned} \\
[/math]

Spero sia sufficientemente chiaro. ;)