Theory of Intelligence • First-Principles AI & STEM Education

Step-by-Step Solution

Step 1: Analyze the Argument of the Absolute Value

Notice that for all real numbers $x \in \mathbb{R}$ :

x^2 \ge 0 \implies x^2 + 1 \ge 1 > 0

Because the expression inside the absolute value bars is strictly positive for every real $x$ , the absolute value bars have no effect:

|x^2 + 1| = x^2 + 1

Step 2: Solve the Simplified Algebraic Equation

Equating the simplified expression to $5$ :

\begin{aligned} x^2 + 1 &= 5 \\ x^2 &= 4 \\ x &= \pm\sqrt{4} = \pm 2 \end{aligned}

Step 3: Graphical Interpretation

The parabola $y = x^2 + 1$ lies entirely above the $x$ -axis. It intersects the horizontal line $y = 5$ at two symmetric points: $(-2, 5)$ and $(2, 5)$ .

Step-by-Step Solution

Step 1: Analyze the Argument of the Absolute Value

Step 2: Solve the Simplified Algebraic Equation

Step 3: Graphical Interpretation

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