Problem 6 - Precalculus

Question

Evaluate each expression. $$|-6+3|$$

Step-by-Step Solution

Answer

The expression evaluates to 3.

Step 1: Simplify Inside the Absolute Value

Begin by simplifying the expression inside the absolute value. You have the expression \(-6 + 3\). Calculate this addition: \(-6 + 3 = -3\).

Step 2: Apply the Absolute Value

The result from Step 1 is \(-3\). Now, take the absolute value of \(-3\). By definition, the absolute value of a number is its distance from zero on the number line, which is always positive. Therefore, \(|-3| = 3\).

Step 3: Conclude the Evaluation

The expression \(|-6+3|\) evaluates to \(3\), as determined in the previous steps.

Key Concepts

Integer Arithmetic

In the world of mathematics, exploring integer arithmetic is like learning a universal language that helps solve problems with whole numbers. Integer arithmetic involves addition, subtraction, multiplication, and division of integers. Integers include positive numbers, negative numbers, and zero. For example, in the problem \(-6 + 3\), we are dealing with addition in integer arithmetic.
Understanding integer arithmetic follows a few simple rules. When you add a positive and a negative integer, the result will take the sign of the number with the larger absolute value. For example:

• If you add \(5 + (-2)\), the result is \((+3)\) because \(5\) is larger than \(-2\).
• If you add \(-6 + 3\), like in our exercise, the result is \(-3\) because \(-6\) has a larger absolute value.

To become proficient in integer arithmetic, it's important to practice using these principles, ensuring you're familiar with both addition and subtraction of whole numbers.

Number Line

A number line is a visual tool in mathematics that helps us understand numbers in a linear format. It is a straight horizontal line where numbers increase as we move to the right and decrease as we move to the left. Let's use our exercise example to understand how a number line helps visualize calculations.
Consider the expression \(-6 + 3\).To perform this calculation, start at \(-6\) on the number line. Since we're adding \(3\), move three steps to the right. This brings us to \(-3\).
Number lines are not only helpful for simple arithmetic but also for visualizing the concept of absolute value. The absolute value of a number, as defined, is its distance from zero on the number line. Thus:

• The absolute value of \(-3\) is \(3\) because it's three steps away from zero.
• Similarly, the absolute value of \(3\) is also \(3\), illustrating that absolute value ignores direction.

Using a number line to illustrate these concepts can greatly enhance your understanding, making abstract arithmetic problems more tangible.

Evaluating Expressions

Evaluating expressions involves simplifying and solving mathematical phrases or equations to reach a final answer. Understanding this process requires knowledge of both integer arithmetic and the concept of absolute value. Let's walk through evaluating the expression \(|-6 + 3|\).First, solve the expression inside the absolute value bars, \(-6 + 3\). Using integer arithmetic, as previously explained, gives \(-3\). Next, apply the absolute value. Absolute value measures how far a number is from zero, which turns negative results into positive ones, meaning \(|-3| = 3\).When evaluating expressions:

• Simplify inside any grouping symbols, like parentheses or absolute value bars, first.
• Apply arithmetic operations according to the proper order, considering all integer rules.
• Convert the outcome into absolute terms if required, like in absolute value expressions.

Evaluating mathematical expressions step by step ensures accuracy and fosters a deeper understanding of the calculation methods used.