Problem 78 - Prealgebra

Question

Simplify the given expression. \(13.65-\mid 11.55-(-4.44)\)

Step-by-Step Solution

Answer

The expression simplifies to \(-2.34\).

Step 1: Understand the Absolute Value

The expression contains an absolute value function: \( \mid 11.55 - (-4.44) \mid \). Absolute value means the distance from zero on a number line, disregarding the sign, so always outputting a non-negative number.

Step 2: Simplify Inside the Absolute Value

First, simplify the expression inside the absolute value: \( 11.55 - (-4.44) \). Subtracting a negative is the same as adding the positive, so \( 11.55 + 4.44 = 15.99 \).

Step 3: Evaluate the Absolute Value

Now, evaluate the absolute value: \( \mid 15.99 \mid = 15.99 \), since the absolute value of a positive number is the number itself.

Step 4: Substitute Back and Simplify the Expression

Substitute the value back into the original expression: \( 13.65 - 15.99 \). Calculate the difference: \( 13.65 - 15.99 = -2.34 \).

Key Concepts

Understanding Absolute Value

Absolute value is a fundamental concept that represents the distance of a number from zero on a number line. This means the value is always non-negative, regardless of whether the original number was positive or negative. For example, the absolute value of both -5 and 5 is 5.

• The symbol for absolute value is two vertical bars, like this: \( \mid x \mid \).
• Absolute values remove any negative sign; hence, \( \mid -3 \mid = 3 \).

In the context of the exercise, when you encounter \( \mid 11.55 - (-4.44) \mid \), you're first tasked with simplifying the expression inside the absolute value brackets. Then, you simply consider how far the resulting number is from zero, which is always a positive or zero.

Performing Addition and Subtraction

Addition and subtraction involving numbers is the process of calculating total sums or differences. This operation is vital in simplifying expressions. When working with these operations:

• Remember that subtracting a negative number is the same as adding its positive counterpart.
• Always handle operations within brackets or inside absolute values before addressing other parts of the expression.

In our example, within the absolute value function, the operation \( 11.55 - (-4.44) \) requires you to turn the subtraction of a negative to an addition: \( 11.55 + 4.44 = 15.99 \). Then, as one often encounters through mathematical problems, you use this result to substitute back into larger expressions for final simplifications.

Working with Negative Numbers

Negative numbers are numbers less than zero and are represented with a "-" sign. Understanding how they interact in mathematical operations is crucial:

• Adding a negative number is akin to subtraction. For example, \( 6 + (-2) \) is the same as \( 6 - 2 \).
• When negative numbers are subtracted, like in \( a - (-b) \), it turns into addition (\( a + b \)).

In this exercise, we see the transition from \( -( -4.44 ) \), which becomes \( +4.44 \). Recognizing this pattern makes handling expressions with negative numbers efficient and intuitive, especially when simplifying larger algebraic equations.

Using a Number Line

A number line is a visual representation that helps understand integers, particularly in evaluating operations like absolute value. Imagine a straightforward line that starts at zero in the middle, with positive numbers extending to the right and negative numbers to the left.

• It helps visualize absolute value, as it is the "distance" a number is from zero.
• When adding or subtracting numbers, moving left represents subtraction, and moving right denotes addition.

For instance, on a number line, subtracting \( 11.55 - (-4.44) \) means you're moving left to subtract, but because it's a double negative, you move \( 4.44 \) steps to the right instead. Number lines are simple yet powerful tools that visually reinforce mathematical concepts, especially for students just familiarizing themselves with algebra.