Problem 1 - Algebra and Trigonometry

Question

The equation \(|X|=3\) has the two solutions____and____

Step-by-Step Solution

Answer

The solutions are 3 and -3.

Step 1: Understand Absolute Value Equation

The equation \(|X| = 3\) represents the absolute value of X being equal to 3. Absolute value gives us the distance of a number from zero on the number line, regardless of direction, which means X can be either positive or negative 3.

Step 2: Set Up Possible Solutions

To solve \(|X| = 3\), consider two possibilities because absolute value can reflect two different scenarios: - X could be positive 3, leading to the equation X = 3. - X could also be negative 3, leading to the equation X = -3.

Step 3: Write Down the Solutions

Based on the scenarios:- If X = 3, then X is positive 3.- If X = -3, then X is negative 3.Thus, the solutions to the equation \(|X| = 3\) are X = 3 and X = -3.

Key Concepts

Solutions to Absolute Value Equations

Solving absolute value equations involves finding all possible values of the variable that satisfy the equation. In the case of the equation \(|X| = 3\), it demonstrates two potential solutions for X. This is because absolute value indicates the number's distance from zero on the number line, which can be either in the positive or negative direction.

To address this problem, we consider:

• First, the positive potential where \(X = 3\).
• Second, the negative potential where \(X = -3\).

Both solutions satisfy the equation because the absolute value of both 3 and -3 is 3. Thus, the equation \(|X| = 3\) leads to the solutions X = 3 and X = -3 respectively. It is crucial to remember that absolute value equations may have two solutions because it reflects all values with that specific distance from zero.

Interpreting Absolute Value

The interpretation of absolute value is fundamental to understanding these equations. Absolute value, denoted as \(|X|\), refers to the distance between a number and zero on the number line. This measure does not differentiate between positive and negative values.

For example, both +3 and -3 are 3 units away from zero. Therefore, the absolute value of both is 3.

• Absolute value disregards the sign, always rendering the result as a non-negative number.
• It often represents real-world scenarios that include distance and magnitude without direction.

Remember that finding absolute values involves simplifying mathematical expressions to determine their distance from the origin, making it a useful tool in various applications.

Properties of Absolute Value Equations

Absolute value equations have several useful properties that help simplify and solve them. Familiarity with these properties helps in making accurate interpretations.

Key properties include:

• **Non-negative outputs**: The result of an absolute value is always non-negative. This is because distance can't be negative.
• **Symmetry about zero**: Both positive and negative inputs result in the same positive output, i.e., \(|a| = |-a|\).
• **Triangle Inequality**: This states that for any real numbers a and b, \(|a + b| \leq |a| + |b|\).

These properties contribute to solving absolute value equations efficiently. Recognizing that solutions involve considering both positive and negative values of the absolute expression reflects a key characteristic of such equations.