Example 15. For all solve the inequality
Solution
Solution. Inequality (20) under the condition is equivalent to the combination of two systems:
\left\{\begin{array} { l }
{ a \geqslant 0 } \\
{ x + 1 \geqslant 0 } \\
{ a ^ { 2 } - x ^ { 2 } > ( x + 1 ) ^ { 2 } , }
\end{array} \quad \left\{\begin{array}{l}
a \geqslant 0 \\
-a \leqslant x \leqslant a \\
x+11$ has a solution $-a \leqslant x1 / \sqrt{ } \overline{2}$. Therefore, system (21) is equivalent to the system
\left\{\begin{array}{l}
x \geqslant-1 \\
a>\sqrt{\overline{2}} / 2 \\
\frac{-1-\sqrt{2 a^{2}-1}}{2}1-1 \leqslant x1-a \leqslant x1$.
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