Problem 114 - Prealgebra and Introductory Algebra

Question

Simplify. $$42-(-30)-65-(-11)$$

Step-by-Step Solution

Answer

The simplified result of the given expression is 7.

Step 1: Understanding Negative Signs

In mathematics, a double negative turns into a positive. So, the expression becomes \(42 + 30 - 65 + 11\).

Step 2: Addition and Subtraction

Now, complete the addition and subtraction from left to right according to the order of operations. The result is \(72 - 65 = 7\).

Key Concepts

Negative Numbers

Negative numbers can sometimes be tricky, but they are just as important to understand as positive numbers. In mathematics, the minus sign (-) is used to denote a negative number. When graphically placed on the number line, negative numbers are to the left of zero. Understanding how to manipulate these numbers requires a familiarity with some basic rules.
When you see two negative signs together, as in the expression \( -(-30) \), these signs 'cancel' each other out, resulting in a positive number. This is due to the fact that subtracting a negative is equivalent to adding the opposite. For instance, in our exercise, \(-(-30)\) becomes \(+30\), and \(-(-11)\) becomes \(+11\). Recognizing this pattern is crucial for simplifying expressions involving negative numbers.

Addition and Subtraction

When dealing with addition and subtraction, especially with negative numbers, it's important to be precise about the order and grouping of numbers. The key to mastering these operations is practice and understanding their properties.
First, determine which numbers you need to add and which ones to subtract. A change in sign, as seen in our example, can turn a subtraction into an addition. Our expression became \(42 + 30 - 65 + 11\) after simplifying the double negatives.

• Adding positive numbers is straightforward: sum them as usual.
• For adding negative numbers, add their absolute values and keep the negative sign.
• For subtracting, consider the subtraction of a number as adding its additive inverse. For example, instead of \(42 - 65\), think \(42 + (-65)\).

In our exercise, once we simplified the signs, we applied these principles orderly to attain the final result of 7.

Order of Operations

In mathematics, order of operations is crucial in solving expressions correctly. The order is often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)). This rule helps in determining the sequence in which mathematical operations should be performed.
However, in our exercise, only addition and subtraction are involved, which simplifies the task. Operations can be handled from left to right as they appear. It is important to keep this order, as doing so ensures accuracy in your calculations. For expressions like \( 42 + 30 - 65 + 11 \), starting from the left:

• First, perform \(42 + 30\) which equals 72.
• Next, subtract \(65\) from \(72\) to get 7.

By sticking to this streamlined approach, mathematical problems are solved both accurately and efficiently.