Theory of Intelligence • First-Principles AI & STEM Education

Step-by-Step Solution

Step 1: Set Up the Double Inequality

Using the bounded absolute value property $|A| < c \iff -c < A < c$ :

-4 < 2x^2 - 3x - 5 < 4

This splits into two simultaneous quadratic inequalities that must both be satisfied.

Step 2: Solve Inequality 1 (Right Part: $< 4$ )

\begin{aligned} 2x^2 - 3x - 5 &< 4 \\ 2x^2 - 3x - 9 &< 0 \end{aligned}

Factor the quadratic:

(2x + 3)(x - 3) < 0

The roots are $x = -\frac{3}{2} = -1.5$ and $x = 3$ . Since the leading coefficient $2 > 0$ , the quadratic is negative strictly between the roots:

x \in \left(-\frac{3}{2}, \; 3\right)

Step 3: Solve Inequality 2 (Left Part: $> -4$ )

\begin{aligned} 2x^2 - 3x - 5 &> -4 \\ 2x^2 - 3x - 1 &> 0 \end{aligned}

Find the roots of $2x^2 - 3x - 1 = 0$ using the quadratic formula:

x = \frac{-(-3) \pm\sqrt{(-3)^2 - 4(2)(-1)}}{2(2)} = \frac{3 \pm\sqrt{9 + 8}}{4} = \frac{3 \pm\sqrt{17}}{4}

Numerically:

r_1 = \frac{3 - \sqrt{17}}{4} \approx \frac{3 - 4.123}{4} \approx -0.281

r_2 = \frac{3 + \sqrt{17}}{4} \approx \frac{3 + 4.123}{4} \approx 1.781

Because the parabola opens upwards, it is strictly positive outside the roots:

x \in \left(-\infty, \; \frac{3 - \sqrt{17}}{4}\right) \cup \left(\frac{3 + \sqrt{17}}{4}, \; \infty\right)

Step 4: Intersect the Two Solution Sets

We take the intersection of the bounded interval $\left(-\frac{3}{2}, 3\right)$ with the exterior rays:

x \in \left(-\frac{3}{2}, \; \frac{3 - \sqrt{17}}{4}\right) \cup \left(\frac{3 + \sqrt{17}}{4}, \; 3\right)

In decimal approximation:

x \in (-1.5, \; -0.281) \cup (1.781, \; 3.0)

Step-by-Step Solution

Step 1: Set Up the Double Inequality

Step 2: Solve Inequality 1 (Right Part: <4< 4<4)

Step 3: Solve Inequality 2 (Left Part: >−4> -4>−4)

Step 4: Intersect the Two Solution Sets

Discussion & Community

Step 2: Solve Inequality 1 (Right Part: $< 4$ )

Step 3: Solve Inequality 2 (Left Part: $> -4$ )