Problem 1 - Algebra 2 and Trigonometry

Question

Explain why the solution set of the equation \(12-|x|=15\) is the empty set.

Step-by-Step Solution

Answer

The equation has no solution because \(|x|\) cannot be negative, making the solution set empty: \(\emptyset\).

Step 1: Understand Absolute Value

The absolute value function \(|x|\) represents the distance of the number \(x\) from zero on the number line. Therefore, \(|x|\) is always non-negative, meaning it is always zero or positive.

Step 2: Analyze the Given Equation

The equation we need to solve is \(12 - |x| = 15\). Our goal is to determine if there exists a real number \(x\) that satisfies this equation.

Step 3: Rearrange the Equation

Rearrange the equation to isolate \(|x|\). We do this by subtracting 12 from both sides: \(12 - |x| = 15\) becomes \(-|x| = 3\), and then multiply both sides by -1 to get \(|x| = -3\).

Step 4: Identify the Contradiction

Observe that the equation now states \(|x| = -3\). However, as established in Step 1, the absolute value \(|x|\) is never negative. This means \(|x| = -3\) is impossible and represents a contradiction.

Step 5: Conclude with the Solution Set

Since there is no value of \(x\) that can satisfy the equation \(12 - |x| = 15\) without violating the properties of absolute value, the solution set is empty. Symbolically, this is represented as the empty set: \(\emptyset\).

Key Concepts

Absolute Value Properties

The absolute value of a number, represented as \(|x|\), measures the distance between that number and zero on the number line. This value is always non-negative, which means it cannot be less than zero. The non-negativity rule is crucial for understanding equations that involve absolute values. Here are the key properties:

• \(|x| = x\) if \(x\) is a positive number or zero.
• \(|x| = -x\) if \(x\) is negative, because taking away the negative sign makes it positive.
• Absolute values are always zero or positive; \(|x|\) can never be negative.

These properties are central when solving equations like \(12 - |x| = 15\) because they set the expectation that the absolute value itself must always be equal to or greater than zero.

Solution Set Determination

Determining the solution set of an equation is like playing detective with numbers. The solution set includes all the values of \(x\) that make the equation true. For the equation \(12 - |x| = 15\), we need to figure out whether any real number \(x\) satisfies this condition.

Let's rearrange the equation to see if a solution exists. Subtract 12 from both sides to isolate the absolute value: \(12 - |x| = 15\) becomes \(-|x| = 3\). To express \(|x|\) directly, multiply every term by -1: \(|x| = -3\).

Now, apply the absolute value properties: \(|x|\) cannot equal -3 since it's always positive or zero. Hence, no number satisfies the equation, meaning the solution set is empty. In mathematical terms, this emptiness is shown as \(\emptyset\), indicating that there are no possible solutions.

Mathematical Contradiction

In mathematics, a contradiction arises when we reach a statement that cannot logically be true. It's like trying to prove that 2 + 2 equals 5.

For the equation \(12 - |x| = 15\), we were led to \(|x| = -3\). Here lies the contradiction:

• The absolute value \(|x|\) is supposed to be non-negative, representing a distance that is zero or more.
• The equation \(|x| = -3\) contradicts this rule because a distance can never be negative.

This fundamentally violates the nature of absolute values and highlights why no real number can satisfy the original equation. When a mathematical problem results in such a contradiction, it indicates there are no solutions, hence the empty solution set.