1. The problem asks us to find the real values of $x$ that satisfy the inequality $$- |x| - 1 \geq 1.$$\n\n2. First, isolate the absolute value term. Add 1 to both sides:\n$$- |x| - 1 + 1 \geq 1 + 1$$\n$$- |x| \geq 2.$$\n\n3. Multiply both sides by $-1$ to get rid of the negative sign. Remember, multiplying an inequality by a negative number reverses the inequality sign:\n$$\cancel{-1} \times - |x| \leq \cancel{-1} \times 2$$\n$$|x| \leq -2.$$\n\n4. The absolute value $|x|$ is always non-negative, so $|x| \leq -2$ means the absolute value is less than or equal to a negative number, which is impossible.\n\n5. Therefore, there are no real values of $x$ that satisfy the inequality. The solution set is the empty set $\emptyset$.\n\n6. On a number line, this means no points are included.\n\nFinal answer: $$\boxed{\emptyset}.$$