Problem 75 - Prealgebra

Question

Simplify the given expression. \(1 .2+|8.74-16.5|\)

Step-by-Step Solution

Answer

8.96

Step 1: Calculate the Absolute Value

First, we need to calculate the value inside the absolute value symbol: \( 8.74 - 16.5 \). Subtract 16.5 from 8.74: \( 8.74 - 16.5 = -7.76 \).The absolute value of \( -7.76 \) is \( 7.76 \).

Step 2: Perform Addition

Now, add the result from Step 1 to 1.2: \(1.2 + 7.76 = 8.96 \).

Key Concepts

Simplifying Mathematical Expressions

When you come across an expression that involves multiple operations, the goal is to simplify it. Simplification means transforming the expression into its simplest form.
This process often involves following a certain sequence to solve operations, which makes it easier to understand and interpret the expression's value.
For example, in the expression \(1.2 + |8.74 - 16.5|\), you must first handle the operations inside the absolute value before proceeding to addition.
By simplifying, we make complex problems easier to deal with and often more aesthetically pleasing, removing unnecessary components, which provides clarity.

• Simplification often helps in comparing and evaluating expressions.
• It assists in identifying properties and patterns hidden in more complex formulations.

Understanding Addition in Expressions

Addition is one of the basic operations in mathematics. It's used to combine numbers to get a sum. In an algebraic expression like our problem, addition is crucial in reaching the correct answer.
Once the expression inside the absolute value is simplified, addition is straightforward.
For example, in the problem, after finding that the absolute value is 7.76, you then add it to 1.2, which gives 8.96.
Understanding addition involves:

• Knowing how to align numbers correctly when performing operations manually.
• Being careful with decimals to maintain accuracy.

In some expressions, addition is not just adding singular numbers but might require combining like terms, especially in polynomials.

Expressions and Absolute Value

Expressions can include numbers, variables, and operators that represent a value. In math, handling expressions often involves solving or simplifying them.
An absolute value, denoted by vertical bars \,|\, is a special expression that always yields a non-negative result.
It signifies the distance a number is from zero, without considering direction.
For instance, the absolute value of \(-7.76\) in the expression \( |8.74 - 16.5| \) results in \(7.76\). This concept is critical because it always returns a non-negative value, simplifying problem-solving.

• The absolute value removes negative signs, which can greatly simplify calculations.
• Understanding this property is essential for solving equations and inequalities involving absolute terms.

Expressions that include absolute values demand careful execution of operations inside the bars before proceeding.