Problem 43 - Algebra A Combined Function

Question

Simplify each expression. \(-10-(-8)+(-4)-20\)

Step-by-Step Solution

Answer

The expression simplifies to -26.

Step 1: Remove Parentheses

Begin by simplifying the parentheses. Notice that a negative sign in front of a parenthesis can change the sign of the number inside it. So, replace \(-(-8)\) with \(+8\). The expression becomes:\[-10 + 8 + (-4) - 20\].

Step 2: Combine Negative Numbers

Add up all the negative numbers together. Here, we have \(-10\) and \(-4)\). Adding these together, we get:\[(-10) + (-4) = -14\].

Step 3: Add Positive Number

Now, add the positive number from the expression, which is \(8\), to the result from the previous step \(-14\). This results in:\[-14 + 8 = -6\].

Step 4: Subtract Last Number

Subtract the remaining term, which is \(20\), from \(-6\). This results in:\[-6 - 20 = -26\].

Key Concepts

Negative Numbers

Negative numbers can be understood as numbers less than zero, and they are often used to represent loss or deficiency. In mathematical terms, a negative number is indicated by a minus sign (-) in front of a numeral, such as \(-5\). When working with negative numbers, it’s important to remember these key points:

• Negative numbers can be added, subtracted, multiplied, and divided, similar to positive numbers. However, their interactions can be a bit different.
• When you subtract a negative number, it's the same as adding its positive counterpart. For example, subtracting \(-8\) is equivalent to adding \(+8\).
• Negative numbers can sometimes cause confusion, especially when parentheses are involved, since the signs can change depending on operations.

These properties are essential when simplifying expressions, as they ensure the correct calculation by accurately managing signs.

Expression Simplification

Expression simplification involves reducing an algebraic expression to its simplest form. This process helps make calculations easier and clearer. In our exercise, the expression starts with a complex set of operations: \(-10-(-8)+(-4)-20\).
To simplify, follow these steps:

• Remove Parentheses: Always eliminate parentheses by appropriately adjusting signs, such as changing \(-(-8)\) to \(+8\).
• Combine Like Terms: Group terms with the same operations (all negatives with negatives, all positives with positives) to simplify calculations effectively.
• Perform Operations: Sequentially carry out addition and subtraction on the grouped terms. This will lead to a simplified numerical result.

Through simplification, the expression \(-10 - (-8) + (-4) - 20\) is handled step by step, removing confusion and focusing only on essential operations. This highlights the importance of sticking to the order of operations to achieve accurate results.

Integer Operations

Integer operations involve basic arithmetic operations (addition, subtraction, multiplication, division) performed on integers. An integer is a whole number that can be positive, negative, or zero. Here are some quick guidelines:

• Addition: When adding integers with the same sign, simply add their absolute values and keep the sign. For different signs, subtract the smaller absolute value from the larger and adopt the sign of the number with the larger absolute value.
• Subtraction: For subtraction, change it to adding the additive inverse (flip the sign of the number being subtracted), then proceed as with addition.
• Multiplication and Division: Multiplying or dividing integers of the same sign gives a positive result, whereas integers with differing signs yield a negative result.

Understanding these operations is crucial for expressions where multiple integers and mixed operations are involved, such as in our expression simplification of \(-10 - (-8) + (-4) - 20\). Mastery of these operations aids in confidently navigating through algebra.