Problem 25 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 25-32: Rewrite the expression without using the absolute value symbol, and simplify the result. $$ |3+x| \text { if } x<-3 $$

Step-by-Step Solution

Answer

For \(x < -3\), \(|3+x| = -3-x\).

Step 1: Understanding Absolute Value

The absolute value of a number represents its distance from zero on the number line, without considering direction. For example, \(|a| = a\) if \(a\geq 0\) and \(|a| = -a\) if \(a < 0\).

Step 2: Determine the Sign of the Expression Inside Absolute Value

Given the expression \(|3+x|\), and the condition \(x < -3\), calculate \(3 + x\). Since \(x < -3\), any value of \(x\) will make \(3 + x < 0\).

Step 3: Rewrite the Expression Without Absolute Value

Apply the property of absolute values. Since \(3 + x < 0\) for \(x < -3\), \(|3+x| = -(3+x)\).

Step 4: Simplify the Expression

Simplify \(-(3+x)\) by distributing the negative sign: \( -(3 + x) = -3 - x\).

Key Concepts

Expression Simplification

Simplifying an expression means to break it down into a simpler form without changing its value. In algebra, you often deal with expressions that can initially seem complex. The goal is to express them in a straightforward way. For instance, if you have the expression

• \(-(3 + x)\),
• you can simplify this by distributing the negative sign across the terms inside the parentheses,
• resulting in \(-3 - x\).

This makes the expression easier to understand and work with.
To simplify an expression accurately, you need to observe mathematical operations.
Distribute any terms correctly and combine like terms when possible.
Remember that your aim is not to alter the value but to express it with fewer terms or less complexity.

Negative Values

Negative values are essential in algebra and can change the outcome of an expression. Understanding negative values is crucial because they affect the result when combined with other numbers through operations like addition and subtraction.

• A negative value is any number less than zero, represented by the minus sign (-).
• When a negative value is added to a positive value, the numbers 'cancel out' to some extent, impacting the final sum.

For example, \(-(3 + x)\) can be tricky, especially since

by multiplying by negative one, the whole expression flips signs.

This concept helps you manage expressions involving absolute values, as it dictates how you should handle the numbers when removing the absolute value symbols.
It's important to remember that the absolute value of a negative number will be positive, changing the neighborhood of the solution.

Algebraic Expressions

Algebraic expressions consist of variables, constants, and operators and are the foundation of equations in algebra.
Understanding these components is crucial for solving equations or simplifying expressions. For instance, \(-3 - x\)

• is a simple algebraic expression.
• It combines constants (\(-3\) ) with variables (\(x\) )
• using standard operators like addition or subtraction.

When dealing with expressions like \(|3 + x|\), identifying how each component interacts is the key to solving problems, especially when absolute values and inequalities are involved.
Recognizing which part of the expression represents what ('where the variable is and how it interacts with constant numbers') helps to rewrite and solve them correctly. Whether you're manipulating by solving, simplifying, or transposing them, knowing the role of each element is indispensable for any algebra problem.