Problem 17 - Elementary and Intermediate Algebra

Question

Add. Do not use the number line except as a check. \(10+(-15)\)

Step-by-Step Solution

Answer

10 + (-15) = -5.

Step 1: Identify the Numbers

Recognize the numbers in the problem: 10 and -15. Note that 10 is positive and -15 is negative.

Step 2: Apply the Addition Rule for Opposite Signs

When adding a positive number and a negative number, subtract the smaller absolute value from the larger absolute value. The sign of the result will be the sign of the number with the larger absolute value.

Step 3: Calculate the Absolute Values

The absolute values are |10| = 10 and |-15| = 15.

Step 4: Subtract the Smaller Absolute Value from the Larger

Subtract 10 from 15: 15 - 10 = 5.

Step 5: Assign the Correct Sign

Since the original number with the larger absolute value is -15, the result will have a negative sign: 10 + (-15) = -5.

Step 6: Check Using Number Line (Optional)

Use a number line to verify the solution if needed: Start at 10 and count 15 units to the left (since -15). You will land at -5.

Key Concepts

Absolute Value

The absolute value of a number is its distance from zero on the number line, regardless of direction. It's always a non-negative number. For example, the absolute value of both 10 and -10 is 10. You write the absolute value of a number using vertical bars. So, \(|10| = 10\) and \(|-10| = 10\) are the same in terms of distance from zero.
To solve the problem 10 + (-15), it's important to know the absolute values of each number: \(|10| = 10\) and \(|-15| = 15\). Knowing these helps us apply the rule for adding integers with opposite signs.
Absolute value is a key concept in many areas of mathematics. Understanding this foundational idea will make working with integers easier and more intuitive.

Opposite Signs Addition

When adding two numbers with opposite signs (one positive and one negative), the process involves subtraction rather than straightforward addition. You subtract the smaller absolute value from the larger one, and the result takes the sign of the number with the larger absolute value.
For the problem 10 + (-15):

• Identify the numbers: 10 is positive and -15 is negative.

Next, find the absolute values: \(|10| = 10\) and \(|-15| = 15\).
Subtract the smaller absolute value from the larger absolute value: 15 - 10 = 5.
Since -15 has the larger absolute value, the result takes the negative sign: therefore, 10 + (-15) = -5.
This method ensures that the sum reflects the magnitude and direction indicated by the number with the larger absolute value.

Number Line

A number line is a visual representation of numbers laid out in a straight line. Each point on this line corresponds to a real number. It helps in understanding the addition and subtraction of integers and their relationships.
To check our earlier calculation, we can use the number line:

• Start at 10 on the number line.
• Since you're adding -15, move 15 units to the left (negative direction).
• You arrive at -5, confirming our earlier result.

This visual check using a number line is useful, especially when initially learning these concepts. It helps reinforce what the absolute value and opposite signs addition rules predict.
Number lines are not just for integers. They are foundational tools in many areas, including rational numbers and even in understanding algebraic concepts.