Problem 103 - Contemporary Precalculus

Question

Write the given expression without using absolute values. $$|b-3| \text { if } b \geq 3$$

Step-by-Step Solution

Answer

Question: Rewrite the expression $$|b-3|$$ without using absolute values. Answer: The expression $$|b-3|$$ can be rewritten as a piecewise function: $$ \begin{cases} b-3 & \text{if } b \geq 3 \\ -(b-3) & \text{if } b < 3 \end{cases} $$

Step 1: Identify positive value scenarios

In this case, we are already given the scenario where the value inside the absolute value signs is positive, which is when $$b \geq 3$$. So, when $$b \geq 3$$, the expression is simply $$(b-3)$$. Step 2: Identify negative value scenarios

Step 2: Identify negative value scenarios

Now, we need to find when the expression inside the absolute value signs is negative. In this case, it is when $$b < 3$$. When $$b < 3$$, the expression becomes $$-(b-3)$$. Step 3: Combine both scenarios

Step 3: Combine both scenarios

Since we have the expressions for both scenarios, we can combine them into a piecewise function. The result is: $$ \begin{cases} b-3 & \text{if } b \geq 3 \\ -(b-3) & \text{if } b < 3 \end{cases} $$

Key Concepts

Piecewise Functions

Piecewise functions are mathematical expressions that define a function using different formulas over different intervals. Imagine a road that has speed limits that vary depending on where you are along the path. Similarly, a piecewise function tells you which mathematical rule applies depending on the value of the input variable.

• It consists of intervals expressed as conditions.
• Each interval has a specific expression or formula.

In our exercise, we have a function that behaves differently depending on whether the variable \( b \) is greater than or less than 3. By using piecewise notation, you can clearly see how the function responds to different inputs, making it easier to understand discontinuous or multi-rule functions.

Positive and Negative Scenarios

When dealing with absolute values, it’s key to understand positive and negative scenarios. Absolute values measure the distance a number is from zero, always giving a non-negative result. To rewrite an absolute value expression, we consider both positive and negative cases of the expression inside the absolute value.
- **Positive Scenario:** If \( b \geq 3 \), then \( |b - 3| = b - 3 \). Here, the expression inside the absolute value is already non-negative, so we can safely drop the absolute value symbols.
- **Negative Scenario:** If \( b < 3 \), the expression \( b - 3 \) is negative. To find its absolute value, we multiply by -1: \( |b - 3| = -(b - 3) \). This operation flips the sign, making it positive without changing its magnitude.

Expressions Without Absolute Values

Simplifying expressions without absolute values involves translating each condition into a basic algebraic expression. By doing this, you remove the absolute value bars and represent the number’s absolute distance under varying conditions.

• **For \( b \geq 3 \):** The expression becomes \( b - 3 \).
• **For \( b < 3 \):** The expression transforms into \( -(b - 3) \), or equivalently \( 3 - b \).

This approach aids in graphing and solving equations, as you work with simple, linear expressions instead of dealing with complex absolute value symbols.

Step-by-Step Solution

Solving problems with absolute values using a step-by-step approach helps ensure thorough understanding and accuracy.

Step 1: Identify Conditions

Start by identifying when the expression inside the absolute value is positive or negative. Here, the conditions are \( b \geq 3 \) and \( b < 3 \).

Step 2: Transform the Expression

Convert the absolute value expression based on these conditions:

• For \( b \geq 3 \), simply take \( b - 3 \).
• For \( b < 3 \), use \( -(b - 3) \).

Step 3: Combine and Represent

Finally, compile these transformations into a piecewise function:\[\begin{cases} b-3 & \text{if } b \geq 3 \ 3-b & \text{if } b < 3\end{cases}\]Using these steps enables a clear breakdown of how absolute values can be resolved into separate linear components.