Problem 52 - Beginning and Intermediate Algebra

Question

Find each sum. $$ [-2+(-11)]+[-12+(-2)]+[18+(-6)] $$

Step-by-Step Solution

Answer

-15

Step 1 - Simplify the first pair

Simplify the first pair of numbers inside the brackets: -2 + (-11). Adding two negative numbers is equivalent to adding their absolute values and putting a negative sign: -2 + (-11) is -13.

Step 2 - Simplify the second pair

Simplify the second pair of numbers inside the brackets: -12 + (-2). Similarly, -12 + (-2) is -14.

Step 3 - Simplify the third pair

Simplify the third pair of numbers inside the brackets: 18 + (-6). Here, subtract 6 from 18, giving: 18 + (-6) = 12.

Step 4 - Add the simplified results

Now, add the simplified results from the previous steps: -13 + (-14) + 12. Start by adding -13 and -14: -13 + (-14) = -27. Then add 12 to -27: -27 + 12 = -15.

Key Concepts

negative numbers

Negative numbers are numbers that are less than zero, and they are often represented with a minus sign (-). They are the opposite of positive numbers. Imagine a number line, zero is in the center, numbers to the right are positive, and numbers to the left are negative. Negative numbers can be tricky because they involve direction. For example, if you owe someone \(5, you could think of it as -5 dollars. It's important to understand how to work with negative numbers to solve problems like the one in the exercise.

When adding negative numbers:

• Think of combining debts. For instance, -2 + -11 is like owing \)2 and then owing \(11 more. Altogether you owe \)13.
• Another way to see it is to add their absolute values and then put a negative sign in front of the result.

Understanding these concepts can help simplify operations with negative numbers.

absolute values

The absolute value of a number is its distance from zero on a number line, regardless of direction. It’s always a non-negative number. We use the absolute value to simplify the addition of negative numbers.

To find the absolute value, you:

• Strip away the negative sign if there is one.
• For example, the absolute value of -11 is 11, because -11 is 11 units away from 0 on the number line.
• Similarly, the absolute value of 6 is 6.

In the context of the original exercise, when faced with -2 + (-11), you can think of combining their absolute values (2 and 11), giving 13. Then, append a negative sign to get -13. This helps in understanding why -2 + (-11) results in -13.

addition

Addition is the process of combining two or more numbers to get a total. It sounds simple, but things get more complex with negative numbers. Let’s break it down for the current exercise:

Step by step:

• First, combine -2 and -11 by adding their absolute values: 2 + 11 = 13. Then, because both numbers are negative, the result is -13.
• Next, combine -12 and -2 the same way: the absolute values 12 and 2 add up to 14. Hence, the result is -14.
• For 18 and -6, think of subtracting 6 from 18, resulting in 12.

So far we have -13, -14, and 12.

Now, adding these results together:

• Start with -13 and -14: Add their absolute values (13 + 14 = 27), then append a negative sign to get -27.
• Finally, add 12 to -27: Think of this as subtracting 12 from 27, which gives us -15.

This step-by-step approach helps you understand how to handle addition, especially involving negative numbers and their absolute values.