Problem 69 - Intermediate Algebra

Question

In the following exercises, simplify. $$ 10-3|9-3(3-1)| $$

Step-by-Step Solution

Answer

7

Step 1 - Evaluate Inside the Parentheses

First, solve the expression inside the innermost parentheses:\(3 - 1 = 2\).

Step 2 - Multiply Inside the Absolute Value

Next, multiply the result by 3 inside the absolute value:\(3(2) = 6\).

Step 3 - Subtract Inside the Absolute Value

Then, subtract this result from 9 within the absolute value bars:\(9 - 6 = 3\).

Step 4 - Evaluate the Absolute Value

Take the absolute value of the result:\(|3| = 3\).

Step 5 - Subtract from 10

Finally, subtract the result from 10:\(10 - 3 = 7\).

Key Concepts

Order of Operations

When simplifying algebraic expressions, the sequence in which you solve parts of the expression is crucial. This sequence is known as the order of operations, and it helps ensure that everyone simplifies expressions in the same way. Remember the acronym PEMDAS to recall the order:

• P: Parentheses
• E: Exponents (including roots, such as square roots)
• M: Multiplication
• D: Division
• A: Addition
• S: Subtraction

Notice that multiplication and division are on the same level; you solve them from left to right. The same goes for addition and subtraction.
In our exercise, notice we started with parentheses first, moving inwards. Following PEMDAS ensures that complex expressions are simplified correctly each time.

Absolute Value

Absolute value represents the distance of a number from zero on a number line, regardless of direction. It’s always a non-negative number.
The absolute value of a number is denoted with vertical bars. For example, \(|-3| = 3\) and \( |3| = 3 \).
In the exercise, we dealt with an absolute value expression \(|9 - 3(3-1)|\). Absolute values make everything inside their bars positive. For instance, if the result inside the bars was negative, we'd turn it to positive. This is essential for correctly simplifying expressions involving absolute value.

Parentheses in Arithmetic

Parentheses (and brackets) help group parts of an expression that should be simplified first. They prevent ambiguity by clarifying which operations should be performed prior to others. Simplifying inside parentheses is the initial step due to the order of operations.
In our example, we had to simplify inside the innermost parentheses first \(3 -1 = 2\). After dealing with the transpose \(:3(2)|\), we handled the content within the absolute value.
Always work from the innermost set of parentheses outward. This means handling operations inside, then proceeding to the next layer. Additionally, this helps in breaking down complex expressions into manageable chunks.