$x^{2}+1\ >\ 4$

Hello Amanita! Let's solve the inequality $x^2 + 1 > 4$ step by step.

Step-by-Step Plan

1. Understand the Problem: We need to find all values of $x$ that make the inequality $x^2 + 1 > 4$ true.
2. Isolate the Variable: Subtract 1 from both sides to simplify the inequality.
3. Solve the Resulting Inequality: Solve the new inequality for $x$.
4. Interpret the Solution: Write the solution in interval notation and explain what it means.

Let's start by isolating the variable.

Step 1: Subtract 1 from both sides

Now, let's solve the inequality $x^2 > 3$.

Step 2: Solve $x^2 > 3$

To solve this, we need to find all $x$ such that the square of $x$ is greater than 3. This happens when $x > \sqrt{3}$ or $x < -\sqrt{3}$.

Let's use the tool to solve $x^2 > 3$ for $x$. Here is the detailed solution, Amanita:

1. We started with the inequality:

2. We subtracted 1 from both sides to isolate the $x^2$ term:

3. To solve $x^2 > 3$, we look for all values of $x$ whose square is greater than 3. This happens in two cases:

   • When $x > \sqrt{3}$
   • When $x < -\sqrt{3}$

   In interval notation, the solution is:

$x \in (-\infty, -\sqrt{3}) \cup (\sqrt{3}, \infty)$

Final Answer:
All values of $x$ such that $x < -\sqrt{3}$ or $x > \sqrt{3}$ will satisfy the inequality $x^2 + 1 > 4$.

If you have any more questions or want to see this on a number line, let me know!