Problem 46 - Algebra A Combined Function

Question

Simplify each expression. \(7-12+(-5)-2+(-2)\)

Step-by-Step Solution

Answer

The expression simplifies to \(-14\).

Step 1: Understand the Problem

The task is to simplify the mathematical expression \(7 - 12 + (-5) - 2 + (-2)\). This involves performing addition and subtraction of various numbers, including negative numbers.

Step 2: Group Positive and Negative Numbers

Start by grouping the positive and negative numbers to make the calculations easier:- Positive: \(7\)- Negative: \(-12, -5, -2, -2\)

Step 3: Calculate the Sum of Negative Numbers

Add up all the negative numbers: \(-12 + (-5) + (-2) + (-2) = -12 - 5 - 2 - 2 = -21\).

Step 4: Add the Positive and Negative Sums Together

Now, add the sum of the positive numbers to the sum of the negative numbers: \(7 + (-21) = 7 - 21 = -14\).

Step 5: Verify the Solution

Recheck the calculations to ensure that there are no errors. The solution to the expression \(7 - 12 + (-5) - 2 + (-2)\) is \(-14\).

Key Concepts

Addition and Subtraction

When it comes to simplifying mathematical expressions, understanding addition and subtraction is key. These operations are complementary, meaning they work together to either increase or decrease values. In simple terms, addition happens when you put numbers together, while subtraction tells you how much one number differs from another.

• Adding numbers increases the total.
• Subtracting numbers decreases the total.

To succeed at simplifying expressions, focus on the numbers and the operations between them. Notice how these two operations interact, especially when multiple numbers are involved. This understanding is crucial when dealing with longer strings of numbers or terms within an expression. By methodically performing each operation, you will arrive at the simplest form of the expression.

Negative Numbers

Negative numbers are just numbers with a minus sign in front. They're like debt. If you owe someone $5, you have -5 dollars. When dealing with negative numbers, it's important to remember a few rules that can make calculations easier.

• Adding a negative number is like subtracting a positive one. For example, adding -3 is the same as subtracting 3.
• Subtracting a negative number can get tricky. It’s like adding the opposite. For instance, subtracting -3 is the same as adding 3.

Negative numbers are common in many expressions, and understanding how they interact with positive numbers is crucial for correctly solving or simplifying those expressions. They can often change the direction or the value of the calculations, so always pay close attention to their role in the equation.

Mathematical Expressions

Mathematical expressions are combinations of numbers, variables, and operations like addition and subtraction. They don’t have an equality sign, unlike equations. Learning to read and simplify them is a vital skill in mathematics. Here’s what to keep in mind:

• Identify each part of the expression first. Look for numbers, operations, and sometimes parentheses.
• Follow the order of operations: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right) – known as PEMDAS.
• Combine like terms, especially when dealing with variables or constants; this simplifies the expression to its most basic form.

Understanding how to break down an expression into manageable parts allows you to effectively approach and simplify it. This skill is essential when tackling progressively more complex mathematical problems.