Iracionální nerovnice

Př.:

x2+2x+5>x−7x2+2x+5≥0D=4−20<0x−7≥0x≥7x∈⟨7;∞)x2+2x+5>x2−14x+4916x>44x>114  ⟹  x∈⟨7;∞)x−7<0x∈(−∞,7)  ⟹  x∈R\begin{align*} \sqrt{x^2 + 2x + 5} &> x - 7 \\[1em] x^2 + 2x + 5 &\ge 0 \\ D = 4 - 20 &< 0 \\[1em] x - 7 &\ge 0 \\ x &\ge 7 \\[0.5em] x &\in \langle7; \infin) \\[0.5em] x^2 + 2x + 5 &> x^2 - 14x + 49 \\ 16x &> 44 \\ x &> \frac{11}{4} \\ \implies x &\in \langle7; \infin) \\[1em] x - 7 &\lt 0 \\ x &\in (-\infin, 7) \\[1em] \implies x &\in \R \end{align*}

Př.:

99x−x2−8≥7x9x−x2−8≥0x2−9x+8≤0(x−1)(x−8)≤0D=⟨1;8⟩7x≥081(9x−x2−8)≥49x2130x2−729x+648≤0x1,2={7265;92}  ⟹  x∈⟨7265;92⟩\begin{align*} 9\sqrt{9x - x^2 - 8} &\ge 7x \\[1em] 9x - x^2 - 8 &\ge 0 \\ x^2 - 9x + 8 &\le 0 \\ (x - 1)(x - 8) &\le 0 \\ D &= \langle1; 8\rangle \\[1em] 7x &\ge 0 \\ 81(9x - x^2 - 8) &\ge 49x^2 \\ 130x^2 - 729x + 648 &\le 0 \\ x_{1, 2} &= \left\{\frac{72}{65}; \frac{9}{2}\right\} \\ \implies x &\in \left\langle\frac{72}{65}; \frac{9}{2}\right\rangle \end{align*}