Problem 111 - Introductory and Intermediate Algebra for College Students 4th

Question

Simplify: \(-9-11+7-(-3) .\) (Section 1.6, Example 3)

Step-by-Step Solution

Answer

-10

Step 1: Simplify within the parentheses

Start by simplifying the expression within the parentheses. Here we have \( -(-3) \), which becomes \( +3 \). This is due to the rule 'minus of a minus becomes a plus'. So, now the expression becomes \( -9 - 11 + 7 + 3 \).

Step 2: Simplify further by performing addition and subtraction

Next, start performing the addition and subtraction in the order they appear. It is important to note that addition and subtraction operations have the same precedence, so they should be performed from left to right. So, the calculation is: \( -9 - 11 + 7 + 3 = -20 + 7 + 3 = -13 +3 \), which simplifies to \( -10 \).

Key Concepts

Order of Operations

Understanding the order of operations is crucial when simplifying algebraic expressions. It tells us the sequence in which math operations should be carried out to ensure consistency and correctness. This sequence is typically remembered by the acronym PEMDAS:

• Parentheses
• Exponents
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

In our exercise, the focus is primarily on parentheses and the operations of adding and subtracting integers. We first deal with what's inside the parentheses. Here, \(-(-3)\) becomes \(+3\), which is a straightforward application of changing two negatives to a positive.

Once parentheses are simplified, we perform addition and subtraction from left to right without any prioritization between them. This ensures we are systematically working through our expression.

Integer Arithmetic

Integer arithmetic involves various operations like addition, subtraction, multiplication, and division, involving whole numbers, negative numbers, and zero. In our exercise, we are primarily dealing with addition and subtraction involving integers:

• When you subtract integers, you are essentially adding their opposite. For example, subtracting \(11\) is the same as adding \(-11\).
• Addition directly sums the integers in the expression.

In the expression \(-9 - 11 + 7 + 3\), we treat each subtraction as adding a negative. Understanding these subtle transformations in arithmetic helps solve complex problems methodically.

Practicing integer arithmetic with various combinations of positive and negative numbers sharpens your skills in handling real-world math applications.

Negative Numbers

Negative numbers are numbers less than zero and are found left of zero on the number line. They are critical in various math operations, particularly in subtraction and when dealing with expressions involving minus signs.In our case, \(-(-3)\) means "take away a negative 3," effectively turning it into a positive 3. Understanding this property—that two negative signs side by side flip to a positive—is essential.

• A plain minus sign stands for a negative number.
• Two consecutive minus signs transform into a plus.

This knowledge helps make expressions like \(-9 - 11 + 7 + 3\) manageable. Easily spotting these transformations is key to simplifying expressions accurately and quickly.

Addition and Subtraction

Addition and subtraction are arithmetic operations with equal precedence and are solved from left to right. When simplifying expressions like the one in our example, you tackle these operations seamlessly, leveraging their properties.Add the positive numbers and subtract the negative ones as they appear.

• Start with \(-9 - 11\), which results in \(-20\).
• Then add 7 to get \(-13\).
• Finally, add 3 to reach the final result of \(-10\).

Remember these steps follow the left-to-right rule specific to addition and subtraction. This approach is straightforward once you recognize that each operation is to be completed before moving to the next term in the sequence.

Consequently, mastering addition and subtraction lays a solid foundation for further math learning, making complex operations more intuitive and straightforward.