Problem 70 - Elementary Algebra

Question

Find the value of each of the following expressions. $$ -(7-11) $$

Step-by-Step Solution

Answer

Answer: 4

Step 1: Evaluate the subtraction inside the parentheses

We first need to evaluate the subtraction inside the parentheses (7-11). Subtract 11 from 7, which gives us -4: $$ 7 - 11 = -4 $$

Step 2: Apply the negation to the result of subtraction

Now, we apply the negation to the result of the subtraction. Applying a negation means multiplying the value by -1: $$ -(-4) = -1 \cdot (-4) $$

Step 3: Multiply the values

Finally, multiply -1 by -4: $$ -1 \cdot (-4) = 4 $$ Therefore, the value of the expression \(-(7-11)\) is 4.

Key Concepts

Integer Operations

Understanding integer operations is crucial when dealing with basic algebra. Integers include whole numbers and their opposites, such as -3, 0, and 4. Operations with integers include addition, subtraction, multiplication, and division. Let's break these down in simple terms:

• Addition: Combine numbers to reach a total. For instance, 3 + 2 equals 5.
• Subtraction: Take one number away from another. Like our example, 7 - 11 is -4.
• Multiplication: Combine repeated additions. -1 times -4 gives 4, as negative times negative is positive.
• Division: Split a number into equal parts. Say 6 divided by 3 equals 2.

Performing these operations accurately sets the foundation for problem-solving in algebra. Practice is key!

Order of Operations

The order of operations is a set of rules that dictate the sequence in which the operations should be performed to correctly solve expressions. Memorized through PEMDAS (Parentheses, Exponents, Multiplication and Division (left-to-right), Addition and Subtraction (left-to-right)), these operations ensure clarity in complex problems. Let's walk through this step-by-step:

• Parentheses: Always solve the operations inside parentheses first. In our case, calculate (7 - 11) which simplifies to -4.
• Negation and Signs: Managed alongside multiplication, treating "-" as multiplying by -1, such as -(-4) becoming 4.

By adhering strictly to these rules, confusion is minimized, and correct answers are more easily attained.

Negative Numbers

Negative numbers represent values less than zero and present unique rules in math operations. Comprehending these rules is essential for accurate calculations. Here's what you need to remember:

• Subtraction: Think of this as reverse addition. Subtracting a negative, like turning 7 - 11 into -4, scales value down.
• Multiplication: When two negatives are multiplied, the result is positive. Therefore, -1 multiplied by -4 results in 4.
• Opposite Values: The opposite of a negative number is positive, such as -(-4) giving 4, highlighting core negation properties.

Grasping the basics of negative numbers makes their use in complex calculations straightforward and logical. Build confidence by continually practicing these concepts.