Problem 85 - Algebra 1

Question

In Exercises \(83-85,\) choose the statement that is true about the given numbers. (A) The number in column A is greater. (B) The number in column B is greater. (C) The two numbers are equal. (D) The relationship cannot be determined from the given information. Column A Column B $$ |x|+8 | x+8 $$

Step-by-Step Solution

Answer

(D) The relationship cannot be determined from the given information.

Step 1: Break down the problem into two cases

The problem must be separated into two distinct cases because the sign of x is essential in determining the relationship. Two cases need to be considered: when \(x \geq -8\) and \(x< -8\).

Step 2: Examine case when x is greater than or equal to -8

If \(x\geq -8\), then \(|x|+8\) equals \(x+8\). Because \(|x|\) equals to \(x\) itself, once \(x\geq -8\). As a result, the number in Column A equals the number in Column B for this case.

Step 3: Examine case when x is less than -8

If \(x< -8\), \(|x|+8\) equals \(-x+8\). This occurs because \(|x|\) equals \(-x\) if \(x<0\). However, the number in Column B, \(x + 8\), is less than the number in Column A, \(-x+8\), for \(x< -8\). As a result, the number in Column A is greater for this case.

Step 4: Bring together the results of each case

The number in Column A will be greater when \(x< -8\), while the two numbers will be equal when \(x\geq -8\). As a result, it's impossible to decide from the given information whether the number in Column A or Column B is greater; it depends on the actual value of x.

Key Concepts

Comparison of Expressions

When comparing mathematical expressions, it's important to understand what each expression represents. In the context of this exercise, we have two expressions: \(|x| + 8\) and \(x + 8\). These might seem similar because they both use the variable \(x\) and contain a constant 8. However, the critical difference lies in the absolute value notation in the first expression.

The absolute value of a number \(|x|\) is the distance of the number from zero on the number line, without considering direction. This means \(|x|\) is always non-negative. Therefore, when we see \(|x| + 8\), we're looking at a non-negative term added to 8. On the other hand, \(x + 8\) doesn't guarantee non-negativity and directly varies with the value of \(x\).

To effectively compare these expressions, we need to consider different values of \(x\), as they affect the expressions differently. Recognizing when each expression winds up greater or equal will be crucial in solving inequality problems with absolute values.

Inequalities

Inequalities express a relationship where one expression might be greater or less than another, typically indicated by symbols such as \(>\), \(<\), or \(=\). For example, the problem we're considering involves a potential inequality between \(|x|+8\) and \(x+8\).

In mathematics, tackling inequalities often involves investigating several scenarios due to the variable's possible values. As inequalities can change based on these values, breaking down the problem into smaller cases (like we do with the sign of \(x\)) can simplify the comparison. This involves not just calculating expressions, but understanding under what conditions one might exceed the other.

• When \(x \geq 0\), both expressions simplify to the same equation, making them equal.
• When \(x < 0\), the absolute value causes a switch, meaning \(|x| = -x\).

Through appropriate manipulation of these expressions, you can determine which is larger or if they are equal based on specific values of \(x\). Understanding the conditions under which each side of the inequality holds provides a fuller comprehension of how to approach problems involving inequalities.

Case Analysis

Case analysis is a methodical technique used to explore different scenarios by breaking down the problem into distinct cases. It's particularly useful when dealing with absolute values or any piece-wise defined function. This approach allows for deeper understanding and ensures all potential outcomes are considered.

In this exercise, case analysis is employed to evaluate the expressions \(|x|+8\) and \(x+8\) under two main scenarios:

• Case 1: When \(x \geq -8\) - Here, \(|x|\) behaves as \(x\), so both expressions simplify to \(x + 8\). Therefore, they are equal.
• Case 2: When \(x < -8\) - In this scenario, \(|x|\) becomes \(-x\) because \(x\) is negative. Consequently, \(|x|+8\) simplifies to \(-x + 8\), which is greater than \(x + 8\).

By examining each case, you can evaluate the equality or inequality within each scenario. Case analysis ensures nothing is overlooked and reveals the nuanced effects the sign of \(x\) has on the expressions. Remember, solving inequalities with absolute values requires careful consideration of each case to capture the full range of possibilities.