Problem 14 - Algebra 2 and Trigonometry

Question

Find the value of each given expression. \(|4-3|+|-1|\)

Step-by-Step Solution

Answer

The value of the expression is 2.

Step 1: Simplify Absolute Expression 1

First, look at the absolute value expression \(|4 - 3|\). Calculate the value inside the absolute value: \(4 - 3 = 1\). The absolute value of 1 is simply 1. So, \(|4 - 3| = 1\).

Step 2: Simplify Absolute Expression 2

Now, evaluate the absolute value of \(-1\), which is written as \(|-1|\). The absolute value of a negative number is the positive equivalent, so \(|-1| = 1\).

Step 3: Add the Results

With both absolute values simplified, add the results from Step 1 and Step 2: \(1 + 1 = 2\). Therefore, the final value of the expression \(|4 - 3| + |-1|\) is 2.

Key Concepts

Understanding Simplifying Expressions

Simplifying expressions involves breaking down mathematical statements into their simplest form. By doing this, it's easier to understand the core of the problem and find the solution. When you see an expression like (|4-3| + |-1|), the goal is to deal with each part separately before combining the results.

Let's start with understanding absolute values. Absolute value refers to the distance of a number from zero on a number line, without considering the direction. This means when simplifying an expression including absolute values, any negative signs are ignored. For example:

• The absolute value of a positive number, such as 4-3 = 1, remains the same.
• The absolute value of a negative number, like |-1|, becomes the positive equivalent, which is 1.

Once individual parts are simplified, they are ready to be combined into a straightforward solution, just as you add numbers normally but with the assurance that you're using their absolute values.

Mastering Step-by-Step Solutions

Step-by-step solutions guide you through a problem in a detailed manner to ensure that every part of the calculation is covered completely. By breaking down a complicated expression into individual steps, you can focus on one element at a time.

Starting with (|4-3| + |-1|), the first step is handling |4-3|. Calculate the inner expression 4-3, which equals 1, and its absolute value is also 1. The next part involves |-1|, turning it into its positive counterpart, which is again 1. Simplifying each absolute expression separately allows you to precisely follow through one part before jumping to the next. Finally, add the simplified values together: 1 + 1 = 2.

• Step 1: Simplify |4-3| to 1.
• Step 2: Simplify |-1| to 1.
• Step 3: Add the results to get 2.

This approach ensures you've understood each component of the problem, making your solution solid and accurate.

Basics of Arithmetic Operations

Arithmetic operations are a fundamental part of mathematics, involving addition, subtraction, multiplication, and division. These operations are the building blocks for simplifying expressions and solving equations.

In the problem (|4-3| + |-1|), the arithmetic operations required are subtraction and addition. First, by using subtraction (4-3), you determine the difference, which is 1. Second, applying addition to combine all parts of the simplified absolute values (1 + 1), resulting in 2.

• Subtraction helps identify the initial expression value.
• Addition finalizes the calculation by combining simplified results.

Understanding these operations is crucial as they lay the groundwork for more complex calculations, ensuring that we handle numbers accurately and efficiently. Applying these concepts in problems involving absolute values aids in reaching precise solutions, such as our final result here, which is 2.