Problem 82 - Intermediate Algebra

Question

In the following exercises, simplify each expression. $$ -58-(-67) $$

Step-by-Step Solution

Answer

The simplified expression is 9.

Step 1: Identify the Expression

The given expression to simplify is ewline -58 - (-67)

Step 2: Simplify the Double Negative

When subtracting a negative number, it is equivalent to adding the positive of that number. Therefore, ewline -58 - (-67) ewline can be simplified to ewline -58 + 67.

Step 3: Perform the Addition

Add the two numbers: ewline -58 + 67. ewline First, find the absolute values, | -58 | = 58 and | 67 | = 67. ewline Then subtract the smaller absolute value from the larger one: ewline 67 - 58 = 9.

Step 4: Determine the Sign of the Result

Since the absolute value of 67 (which is 67) is greater than the absolute value of -58 (which is 58), the result takes the sign of 67, which is positive. ewline Thus, -58 + 67 = 9.

Key Concepts

double negative

In algebra, handling double negatives can be tricky, but they follow simple rules. When you see a double negative, it means you're effectively adding a positive number. For example, in the expression \(-58 - (-67)\), the two negatives next to each other make a positive. So, instead of subtracting, you're adding 67 to -58. Always remember:

• \(-(-x) = +x\)

This can simplify your calculations significantly.

absolute value

Absolute value helps you understand the magnitude of a number, regardless of its sign. It’s written as \( |x| \). For instance, for both -58 and 58, the absolute value is 58. The same rule applies to positive numbers.

• |-58| = 58
• |67| = 67

Absolute value represents how far a number is from zero on the number line, without considering which direction.

integer addition

Adding integers involves paying attention to their signs. If the integers have different signs, subtract the absolute values and take the sign of the number with the larger absolute value. Here’s an example from our exercise:

• Expression: \(-58 + 67\)
• Absolute values: 58, 67
• Subtract: 67 - 58 = 9

Since 67 has the larger absolute value and is positive, the result is a positive 9. Understanding this rule simplifies complex addition problems.

sign of the result

Determining the sign of the result is crucial when simplifying expressions. After computing the absolute values and their difference, the sign comes from the integer with the larger absolute value. In the example problem:

• Expression: \(-58 + 67\)
• Larger absolute value: 67

Since 67 is positive, the result is positive. Hence, \(-58 + 67 = 9\). Whenever you're in doubt, compare the absolute values, perform the subtraction, and attach the sign of the larger one to your answer.