Problem 25 - Precalculus: Functions and Graphs

Question

Rewrite the expression without using the absolute value symbol, and simplify the result. $$|3+x| \text { if } x<-3$$

Step-by-Step Solution

Answer

The expression simplifies to \\(-3 - x\\) for \\(x < -3\\).

Step 1: Understand Absolute Value

The absolute value of a number, represented by \(|a|\), is the distance of that number from zero on a number line. It is always a non-negative number.

Step 2: Analyze the Condition

The condition given is \(x < -3\). When \(x < -3\), \((3 + x) < 0\), meaning \((3 + x)\) is negative.

Step 3: Apply the Absolute Value Rule

If \(y < 0\), then \(|y| = -y\). Since \((3 + x) < 0\), we have \(|3 + x| = -(3 + x)\).

Step 4: Simplify the Expression

Simplifying \(-(3 + x)\) gives \(-3 - x\).

Key Concepts

Understanding Inequalities

Inequalities are a fundamental concept in mathematics that help us compare two values or expressions. They tell us whether one expression is greater than, less than, or equal to another. When we deal with inequalities, we often encounter these symbols:

• \(<\) means "less than"
• \(>\) means "greater than"
• \(\leq\) indicates "less than or equal to"
• \(\geq\) stands for "greater than or equal to"

When solving inequalities, such as \(x < -3\), it's crucial to understand that we are describing a range of values. In this case, any value of \(x\) that is less than \(-3\) will satisfy the inequality. Inequalities help define conditions under which certain mathematical expressions behave in specific ways, which is essential in analyzing expressions like the one in our exercise. By establishing the relationship \(x < -3\), we can determine the nature of the expression inside the absolute value, whether it is positive or negative, and proceed accordingly.

Expression Simplification

Simplifying an expression is the process of reducing it to its simplest form, making it easier to understand and work with. In the exercise, we took \(3 + x\), which is within the absolute value \(|3 + x|\), and recognized it as negative when \(x < -3\).

The rule for absolute values states: if \(y < 0\), then \(|y| = -y\). Applying this rule, we changed \(3 + x\) into \(- (3 + x)\). This results in \(-3 - x\). Simplifying the expression involved:

• Observing and understanding the negative nature of the value due to the inequality.
• Applying appropriate mathematical rules.
• Performing straightforward arithmetic operations to reach the expression \(-3 - x\).

Expression simplification is central to making complex expressions manageable and easier for further analysis or solving. It highlights the importance of operations' order, application of mathematical rules, and careful consideration of initial conditions, such as the value of \(x\) given in the inequality.

Mathematical Reasoning

Mathematical reasoning involves logical thinking and the systematic use of mathematical principles to solve problems. In this exercise, reasoning is crucial for understanding how to transition from the absolute value expression \(|3 + x|\) given \(x < -3\), to the simplified form \(-3 - x\).

Let's break down the reasoning process:

• Firstly, we need to interpret the absolute value correctly as the non-negative distance from zero.
• Next, we analyze the condition \(x < -3\), helping us conclude that \(3 + x\) is negative.
• Based on this analysis, we apply the rule that the absolute value of a negative is its negation, so \(|3 + x| = -(3 + x)\).
• Finally, we simplify the negation to reach \(-3 - x\).

This reasoning helps not only in simplifying expressions but also provides a means of verifying the logical steps taken during solution. Mathematical reasoning ensures that each step follows logically from the previous one, maintaining consistency and correctness throughout the problem-solving process.