Problem 41 - Elementary Algebra

Question

For the following exercises, perform the indicated operations. $$ -12-(-11) $$

Step-by-Step Solution

Answer

$$

Step 1: Write the given expression

Write the given expression as it is: $$-12 - (-11)$$

Step 2: Rewrite the subtraction as addition

Recall that subtracting a negative number is the same as adding its opposite. So, rewrite the expression as: $$ -12 + 11 $$

Step 3: Perform the addition

Now add the numbers \(-12\) and \(11\): $$-12 + 11 = -1$$

Step 4: Final answer

Therefore, $$-12-(-11) = -1$$

Key Concepts

Addition of Integers

Addition of integers is a fundamental concept in mathematics. It involves combining two or more integer values to find their sum. Integers include all whole numbers and their negative counterparts, so this operation covers a broader range than just simple arithmetic with positive numbers. Here's what to keep in mind when adding integers:

• When both integers have the same sign (both positive or both negative), add their absolute values and keep the common sign. For example, \(-5 + (-3) = -8\).
• If the integers have different signs, subtract the smaller absolute value from the larger one, and take the sign of the integer with the larger absolute value. For example, \(-12 + 11 = -1\).

This is because the addition of integers is about finding the resultant value when both positive and negative quantities are combined. It's like taking steps forward and backward on a number line!

Subtraction of Integers

Subtraction of integers can initially seem more complicated than addition, but the key lies in understanding its relationship with addition. Subtraction is essentially the addition of a negative. This might sound confusing at first, but here’s a breakdown:

• To subtract an integer, add its opposite. This means that \(a - b\) is the same as \(a + (-b)\).
• For example, in the exercise \(-12 - (-11)\), we rewrite it as \(-12 + 11\). This transforms the subtraction problem into an addition one, making it simpler to handle.

When you subtract a negative number, it's like taking away the idea of debt, which effectively means adding a positive number. This alteration can help make subtraction of integers easier to visualize and solve.

Negative Numbers

Negative numbers are numbers that are less than zero and are typically represented with a minus sign (\(-\)). Understanding negative numbers is crucial because they appear in various real-life situations, such as temperatures below freezing or debts in financial contexts.

• On a number line, negative numbers are positioned to the left of zero. The further left you go from zero, the "smaller" the number, even though the numerical value might appear larger in magnitude (e.g., \(-20\) is less than \(-5\)).
• Negative numbers follow the same basic arithmetic rules as positive numbers, but with additional rules for operations:

When performing operations with negative numbers, such as in the exercise \(-12 - (-11)\), recognizing that two negatives make a positive helps simplify tasks. With time, working with negatives becomes as intuitive as handling positive numbers.