Problem 60 - Elementary Algebra

Question

For the following exercises, perform the indicated operations. $$ (11-8)-(1-6) $$

Step-by-Step Solution

Answer

Question: Simplify the expression (11 - 8) - (1 - 6). Answer: 8

Step 1: Evaluate the expressions inside the parentheses

We have two expressions inside the parentheses: 1. 11 - 8 2. 1 - 6 Now evaluate each of them: 1. 11 - 8 = 3 2. 1 - 6 = -5 So the given expression becomes: (3) - (-5)

Step 2: Subtract the second expression from the first

Now we need to subtract the second result (-5) from the first result (3): 3 - (-5) = 3 + 5 (because subtracting a negative number is the same as adding its absolute value)

Step 3: Perform the final operation

Finally, add the two numbers: 3 + 5 = 8 So, the simplified expression is: 8

Key Concepts

Evaluating Expressions

When we come across an algebraic expression like \( (11-8)-(1-6) \), our first step is to evaluate the expressions within the parentheses. Evaluating expressions is a process of performing the operations and simplifying the expression to find its value. This is done by following the order of operations, also known as BIDMAS or PEMDAS—Brackets, Indices or Exponents, Multiplication and Division, and Addition and Subtraction.

In our example, we first look at \(11 - 8 = 3\) and \(1 - 6 = -5\). The parentheses indicate that we should work out these smaller calculations before dealing with the rest of the expression. Evaluating expressions requires careful attention to ensure accuracy, particularly when dealing with negative numbers and the operation signs.

Subtracting Negative Numbers

A common point of confusion in algebra is dealing with negative numbers, particularly when they're part of a subtraction operation. The key concept to remember is that subtracting a negative number is the same as adding its positive counterpart. This rule simplifies calculations and helps to avoid errors.

Looking at our example, once we have evaluated the expressions inside the parentheses, we are left with \(3 - (-5)\). Subtracting negative five from three refers to moving five units in the positive direction on the number line, starting from three; so effectively, you are adding five to three. The expression thus becomes \(3 + 5\), which equals 8. Remembering this rule about negative numbers not only makes subtraction easier but also facilitates quicker mental arithmetic and boosts confidence in solving algebraic equations.

Parentheses in Algebra

Parentheses play a critical role in algebraic expressions. They indicate which operations should be performed first and help to clarify the order in which complex calculations are to be undertaken. When you encounter an expression with multiple sets of parentheses, like our example \( (11-8)-(1-6) \), you need to evaluate the expressions within each set of parentheses before doing anything else.

Once you determine the values within the parentheses—\((11 - 8)\) and \((1 - 6)\)—you have created a simpler expression, which is now \(3 - (-5)\). This illustrates the value of parentheses in organizing and simplifying expressions. Without these guiding symbols, there would be ambiguity in the sequence of operations, potentially leading to incorrect answers and greater confusion in algebra. Always respect the bounds set by parentheses for accurate and reliable algebraic solutions.