Problem 11 - Beginning and Intermediate Algebra

Question

For each expression, label the order in which the operations should be performed. Do not actually perform them. $$ 3 \cdot 5-2(4+2) $$

Step-by-Step Solution

Answer

1. Parentheses, 2. Multiplication, 3. Multiplication, 4. Subtraction

Step 1: Identify Multiplication and Division

Identify any multiplication or division in the expression. Here, multiplication is present with 3 and 5: \[ 3 \times 5 \]

Step 2: Identify Parentheses and Perform Operation Inside

Identify and solve any operations inside parentheses first. Here, inside the parentheses is 4 + 2: \[ 4 + 2 \]

Step 3: Perform Multiplication Associated with Parentheses

Multiply the result from the parentheses by 2. Since the operation 2(4+2) indicates multiplication: \[ 2 \times (4 + 2) \]

Step 4: Perform Addition and Subtraction from Left to Right

Finally, look at any addition or subtraction left in the expression. Here, it is 3 \times 5 minus the result of 2(4+2): \[ 3 \times 5 - 2(4+2) \]

Key Concepts

Parentheses

When solving mathematical expressions, the first step is always to handle anything inside parentheses. Parentheses group parts of the expression that should be solved first. This means you perform the operations inside parentheses before anything else.

For example, in the expression \[ 3 \times 5 - 2(4 + 2) \], we first solve \( 4 + 2 \), which equals 6. This clarifies what the inner result will be when we move to the next steps.

Remember, always look for the parentheses and simplify them before considering other operations.

Multiplication and Division

After parentheses, the next operations to consider are multiplication and division. We handle these from left to right as they appear in the expression.

In our example, after evaluating the parentheses, we see \( 3 \times 5 \) and \( 2 \times 6 \) (since 4 + 2 = 6). We perform these calculations:

• \( 3 \times 5 = 15 \)
• \( 2 \times 6 = 12 \)

Focus on doing multiplication and division steps as they appear from left to right in the expression.

Addition and Subtraction

Finally, after handling parentheses and any multiplications or divisions, we perform addition and subtraction operations. These are also processed from left to right in the order they appear.

In our expression, after solving \( 3 \times 5 \) and \( 2 \times (4 + 2) \), we are left with:

• \( 15 - 12 \)

Performing this subtraction gives us \( 3 \).

To summarize:

• Simplify any expressions inside parentheses.
• Perform multiplication and division from left to right.
• Finally, handle addition and subtraction from left to right.

This order helps ensure consistency and accuracy in solving mathematical expressions. Remembering this order will make it easier to tackle even complex problems.