Problem 28 - Precalculus: Functions and Graphs

Question

Rewrite the expression without using the absolute value symbol, and simplify the result. $$|7+x| \text { if } x \geq-7$$

Step-by-Step Solution

Answer

The expression \(|7+x|\) simplifies to \(7+x\) when \(x \geq -7\).

Step 1: Understanding Absolute Value

The absolute value function, denoted as |x|, gives the distance of a number x from zero on the number line. If x is zero or positive, |x| is equal to x. If x is negative, |x| is equal to -x.

Step 2: Analyze the Condition

The condition given is that \(x \geq -7\). This means x is either greater than or equal to -7, which implies that \(7 + x\) is non-negative (since \(7 + x \geq 0\) when \(x \geq -7\)).

Step 3: Rewrite the Expression

Since \(7 + x \geq 0\) for all \(x \geq -7\), the expression \(|7 + x|\) is equal to \(7 + x\). This is because the absolute value of a non-negative number is the number itself.

Step 4: Simplification

The expression \(7 + x\) is already in its simplest form as there are no like terms to combine or factors to simplify. Therefore, \(|7 + x|\) simplifies to \(7 + x\) under the condition \(x \geq -7\).

Key Concepts

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operations. Imagine expressions as a way to represent real-world situations using math. For instance, the expression \(7 + x\) includes a constant (7) and a variable (x).

• Constants are fixed numbers, like 7 in our example.
• Variables are symbols, often letters, that stand for unknown or changeable numbers.

With algebraic expressions, you can plug in different values for the variables to get various outcomes.The expression \(|7 + x|\) involves an absolute value, which acts like a protective wrapper for the expression. It allows us to focus on the non-negative outcome of the manipulation, representing how far a number is from zero regardless of its sign.

Simplification

Simplifying algebraic expressions involves reducing them to their most straightforward form with the least number of operations or terms. In our example, simplifying \(|7+x|\) for \(x \geq -7\) means recognizing the conditions under which we can remove the absolute value symbols.

• When simplifying, always examine any conditions given (like \(x \geq -7\) in this case).
• Determine if the expression is already in its simplest form or if further operations are needed.

For instance, once we determine that \(7 + x \geq 0\) due to the condition \(x \geq -7\), we learn that we can remove the absolute value signs. This is because when \(7 + x\) is non-negative, the simplified result, \(7 + x\), holds true across the given range.It’s a tidy way to reflect the true size of the expression without changing its value due to its conditions.

Piecewise Functions

Piecewise functions are like patchwork quilts of math—they have different expressions for different parts or pieces. They let us handle situations where a single formula doesn’t fit all scenarios. This concept connects to our problem through the condition that \(x\) is greater than or equal to \(-7\).

• Piecewise functions allow for flexibility and can adapt to various situations, using different rules for different intervals.
• Each 'piece' of the function operates only over a specified set of values.

In our case, \(|7+x|\) behaves simply as \(7+x\) when \(x \geq -7\). We only need one piece here because the condition guarantees the expression within the absolute value is always non-negative, simplifying the situation.In problems with piecewise functions, imagining each part as a little separate expression activated under certain conditions can help. This ensures we're always using the right operation for the right set of inputs.