Problem 118 - Introductory Algebra for College Students

Question

$$\text { Simplify: } 3(1-2 \cdot 5)-(-28)$$

Step-by-Step Solution

Answer

The simplified form of the given expression is 1.

Step 1: Deal with the Operation inside the Parentheses

First calculate the operation inside the parentheses: \(3(1-2*5)\). In this case that means calculating \(2*5\), which equals 10, then subtracting this from 1. This results in 1-10=-9, leaving the equation: \(3*-9 - (-28)\).

Step 2: Perform the Multiplication Operation

Next, perform multiplication operation: Multiply 3 by -9 to get -27. Now the equation simplifies to: \(-27 - (-28)\).

Step 3: Deal with Double Negatives

Negating a negative number result in a positive number. So, \(-(-28)\) becomes \(+28\), meaning our equation is now \(-27 + 28\).

Step 4: Perform the Addition Operation

Finally, we add -27 and 28, resulting in 1.

Key Concepts

Understanding the Order of Operations

When simplifying algebraic expressions, following the correct order of operations is crucial. It's like following a recipe in the right sequence to get the perfect dish. The order of operations can be remembered using the acronym PEMDAS which stands for:

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

In our example, the expression is \(3(1-2 \cdot 5)-(-28)\). By applying the order of operations, we first address the parentheses. Multiplication within the parentheses comes before subtraction, so \(2 \cdot 5\) is calculated first. Ensure consistency in following these steps for reliable results every time you simplify an expression.

Working with Integer Operations

Integer operations are fundamental when dealing with numbers, especially in algebra. An integer is a whole number that can be positive, negative, or zero. Understanding how to manipulate these through addition, subtraction, multiplication, and division is vital.

In algebra, the expression often involves various operations with integers. For example, in \(3(1-2 \cdot 5)-(-28)\), once the operations inside the parentheses are solved, you're left with \(-9\). When multiplying by 3, as in the expression \(3 \times -9\), it's important to remember the rules:

• Positive × Positive = Positive
• Negative × Negative = Positive
• Positive × Negative = Negative
• Negative × Positive = Negative

So multiplying 3 by \(-9\) gives us \(-27\). In essential integer operations, negative and positive integers behave in predictable ways, allowing you to simplify expressions accurately.

Multiplication in Algebrics Expressions

Multiplication is an essential operation in algebra and often works hand in hand with other operations such as addition and subtraction. It involves repeatedly adding a number by itself. In different contexts, it can also represent scaling a number.

Returning to our example, once the operations inside the parentheses are solved, we perform multiplication. We have \(3 \times -9\), which results in \(-27\). Remember the rules of multiplication, especially when dealing with negative numbers. Multiplying two numbers with different signs yields a negative result. This understanding is crucial in simplifying expressions correctly, as in our given expression.

Adding and Subtracting in Algebra

Addition and subtraction are the final layers of operations when simplifying expressions, adding depth to your understanding of algebra.

In the expression \(-27 - (-28)\), we must handle the double negatives first. Double negatives transform into positives. Therefore, subtracting a negative number, like \(-(-28)\), effectively means you need to add the positive counterpart. This changes our expression to \(-27 + 28\).

• Adding integers with different signs: Subtract the smaller absolute value from the larger and keep the sign of the number with the larger absolute value.
• In \(-27 + 28\), subtract 27 from 28 to get 1.

Understanding how to manipulate addition and subtraction involving both positive and negative numbers is essential to mastering algebra. It allows you to transition expressions accurately into simplified forms.