Problem 37 - Prealgebra

Question

Simplify as much as possible by first changing all subtractions to addition of the opposite and then adding left to right. $$-8-4-2$$

Step-by-Step Solution

Answer

The simplified expression is \(-14\).

Step 1: Change Subtractions to Addition of Opposites

Rewrite each subtraction as the addition of the opposite. The expression \(-8 - 4 - 2\) can be rewritten as \(-8 + (-4) + (-2)\).

Step 2: Add from Left to Right

First, add \(-8 + (-4)\). This results in \(-12\), because adding two negative numbers involves keeping the negative sign and adding their absolute values.

Step 3: Continue Adding from Left to Right

Now, add \(-12 + (-2)\). This results in \(-14\), because once again, you add the absolute values of the negative numbers and keep the negative sign.

Key Concepts

Adding Integers

When you're dealing with integers in math, particularly in problems requiring simplification, understanding how to add them is crucial. Integers include all whole numbers and their opposites like \(-3, 0, \text{ and } 7\). Adding integers might seem confusing at first, but it's rooted in simply following some straightforward rules:

• Same Sign: If both numbers have the same sign, add their absolute values. The result keeps that sign. For example, \(-2 + (-5) = -7\), because we're adding two negatives together.


• Different Signs: If the numbers have different signs, subtract the smaller absolute value from the larger. The result takes the sign of the number with the larger absolute value. For example, \(7 + (-3) = 4\), because substracting the absolute values \(7 - 3\) retains the sign of 7, which is positive.

This straightforward approach helps in accurately managing integer addition.

Subtraction as Adding Opposites

Conversion of subtraction into addition of opposites is a mathematical trick that simplifies working with integers. When we say subtraction can be seen as adding opposites, it means:

• Opposite Numbers: The opposite of a number is simply a number with the opposite sign. So, the opposite of \(5\) is \(-5\), and the opposite of \(-3\) is \(3\).


• Converting Subtraction: \(a - b\) becomes \(a + (-b)\). For example in the expression, \(12 - 5\), you can rewrite this as \(12 + (-5)\), making it easier to keep track of signs.

By transforming subtraction into this form, you can handle any integer operation just as you would handle integer addition, minimizing errors and connecting the process more intuitively.

Negative Numbers

Negative numbers are essential in mathematics, especially when navigating operations involving integers. They represent numbers less than zero on the number line. Understanding negative numbers includes:

• Position on Number Line: Negative numbers are found to the left of zero on the number line, each step left representing going smaller.


• Significance in Operations: When adding two negative numbers, combine their absolute values and retain the negative sign, as opposed to positive numbers, which follow the usual operations.
• Real-World Contexts: Negative numbers often represent things like debt or a drop in temperature, giving them real-world significance and utility.

Mastering negative numbers empowers students to excel in various mathematical contexts, smoothly transitioning between theoretical and practical applications.