Problem 47 - Beginning and Intermediate Algebra

Question

Add See Examples \(\ell\) through 7 . $$ |5+(-10)| $$

Step-by-Step Solution

Answer

5

Step 1: Determine the Sum Inside the Absolute Value

First, calculate the sum inside the absolute value. We have the expression \(5 + (-10)\). Begin by considering the values of 5 and -10. To find \(5 + (-10)\), subtract 10 from 5, resulting in -5.

Step 2: Apply Absolute Value

Now that we have the sum \(-5\), we need to find its absolute value. The absolute value of a number is the non-negative distance of the number from zero on the number line. Therefore, \(|-5| = 5\).

Key Concepts

Addition of Integers

Adding integers involves combining whole numbers, which can be positive or negative, to find their sum. When you have a situation like \(5 + (-10)\), it means you are adding a positive integer, 5, and a negative integer, -10. To solve this:

• If both numbers are positive, you simply add them as usual.
• If both numbers are negative, add their absolute values, but the sum will be negative.
• If one number is positive and the other is negative, subtract the smaller absolute value from the larger absolute value, and the sign of the larger controls the sign of the result.

In this example, 5 is positive and -10 is negative. So, you subtract 5 from 10, due to the absolute values, to get -5. This helps to see how adding integers works and how different signs affect the result.

Integer Operations

Integer operations are mathematical calculations involving whole numbers. These operations include addition, subtraction, multiplication, and division. When working with integer operations, pay attention to the signs:

• **Addition:** When adding numbers with the same sign, keep the sign, and add the numbers. When adding numbers with different signs, subtract the smaller from the larger and use the sign of the larger absolute value.
• **Subtraction:** This can be thought of as adding the opposite. For instance, subtracting a negative is the same as adding a positive.
• **Multiplication and Division:** When numbers have the same sign, the result is positive. When they have different signs, the result is negative.

For our problem, since we added integers of different signs, we focused on addition principles, determining that the large absolute value dictates the sign of the outcome, -5 before taking the absolute value.

Number Line

A number line is a visual tool that helps in understanding the positioning and operations of numbers in abstract math concepts. Numbers increase as you move to the right and decrease as you move to the left. Negative numbers appear left of zero while positive numbers are on the right.
In working with integers like in \(5 + (-10)\), start at 5 on the number line and move 10 steps to the left because you are adding a negative. You will land at -5. This visualization can help you better grasp why \(5 + (-10) = -5\).
The absolute value, then, is the number of spaces from zero, regardless of direction. This means -5 has an absolute value of 5, making it five steps from zero. Understanding and drawing number lines can solidify your comprehension of integer operations.