The values of satisfying the equation are:

Explanation: 1. Identifying the Core Modulus Property To solve this equation efficiently, let's substitute the complex algebraic expressions with simpler variables. Let: Now substitute and back into the original given equation: According to the fundamental properties of absolute values, the property holds true if and only if: and have the same sign (or is zero), meaning . The absolute value of is greater than or equal to the absolute value of , meaning . Equivalently, this condition simplifies directly to: Since , we know that for any real value of , , which means . Since is strictly positive ( ), for the condition to hold, must be non-negative and greater than or equal to : 2. Setting up the Inequality Substitute the expressions for and back into the simplified condition : Move all terms to the left-h

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AMU 2026 — Mathematics PYQ

AMU | Mathematics | 2026

The values of $x$ satisfying the equation $∣ (x^{4} - 4) - (x^{2} + 2) ∣ = ∣ x^{4} - 4∣ - ∣ x^{2} + 2∣$ are:

Choose the correct answer:

Explanation

1. Identifying the Core Modulus Property

To solve this equation efficiently, let's substitute the complex algebraic expressions with simpler variables. Let:

$a = x^{4} - 4$

$b = x^{2} + 2$

Now substitute $a$ and $b$ back into the original given equation:

$∣ a - b ∣ = ∣ a ∣ - ∣ b ∣$

According to the fundamental properties of absolute values, the property $∣ a - b ∣ = ∣ a ∣ - ∣ b ∣$ holds true if and only if:

$a$ and $b$ have the same sign (or $a$ is zero), meaning $a \cdot b \geq 0$ .
The absolute value of $a$ is greater than or equal to the absolute value of $b$ , meaning $∣ a ∣ \geq ∣ b ∣$ .

Equivalently, this condition simplifies directly to:

$a \cdot b \geq 0 and ∣ a ∣ \geq ∣ b ∣$

Since $b = x^{2} + 2$ , we know that for any real value of $x$ , $x^{2} \geq 0$ , which means $b \geq 2$ . Since $b$ is strictly positive ( $b > 0$ ), for the condition to hold, $a$ must be non-negative and greater than or equal to $b$ :

$a \geq b$

2. Setting up the Inequality

Substitute the expressions for $a$ and $b$ back into the simplified condition $a \geq b$ :

$x^{4} - 4 \geq x^{2} + 2$

Move all terms to the left-hand side to form a quadratic-style polynomial inequality:

$x^{4} - x^{2} - 4 - 2 \geq 0$

$x^{4} - x^{2} - 6 \geq 0$

3. Factoring the Polynomial

Let $y = x^{2}$ . The inequality turns into a standard quadratic equation layout:

$y^{2} - y - 6 \geq 0$

Factor the quadratic equation by splitting the middle term:

$y^{2} - 3 y + 2 y - 6 \geq 0$

$y (y - 3) + 2 (y - 3) \geq 0$

$(y - 3) (y + 2) \geq 0$

Now, substitute $x^{2}$ back in place of $y$ :

$(x^{2} - 3) (x^{2} + 2) \geq 0$

4. Finding the Critical Intervals

Let's analyze the two factored components:

The term $(x^{2} + 2)$ is always strictly positive ( $x^{2} + 2 \geq 2$ ) for all real values of $x$ .
Because $(x^2 + 2) > 0$ , we can divide both sides of the inequality by it without changing the direction of the inequality sign:

$x^{2} - 3 \geq 0$

$x^{2} \geq 3$

$∣ x ∣ \geq 3$

This absolute value inequality expands into the following intervals on a number line:

$x \leq - 3 or x \geq 3$

Expressed in standard interval notation:

$x \in (- \infty, - 3] \cup [3, \infty)$