Problem 56 - Beginning and Intermediate Algebra

Question

Add See Examples \(\ell\) through 7 . $$ -3.7+[0.1+(-0.6)+8.1] $$

Step-by-Step Solution

Answer

The result of the expression is 3.9.

Step 1: Simplify Inside the Brackets

First, focus on simplifying the expression inside the brackets: \([0.1 + (-0.6) + 8.1]\).Start by adding the first two terms: \[0.1 + (-0.6) = -0.5\] Then add the result to the third term:\[-0.5 + 8.1 = 7.6\]. So the expression inside the brackets simplifies to 7.6.

Step 2: Substitute the Simplified Expression

Now that we know that the expression inside the brackets is 7.6, substitute it back into the original expression: \[-3.7 + 7.6\].

Step 3: Compute the Final Expression

Perform the addition: \[-3.7 + 7.6\].To do this, think of moving from -3.7 up the number line 7.6 units. This results in,\[3.9\].

Key Concepts

Addition of decimals

Decimal addition might seem a bit tricky at first, but it is quite similar to adding whole numbers. When we add decimals, it's crucial to line up the decimal points. This ensures each digit is in the correct place value row: ones under ones, tenths under tenths, and so on. For instance, in the original exercise, one of the calculations required was to add 0.1 and -0.6. When we align the decimals like this:

• 0.1
• - 0.6

it's clear where each digit ought to be. When adding, it helps to regroup the numbers, especially when one number is small and the other is negative. This is what was done when finding 0.1 + (-0.6) = -0.5. Just make sure to take care when moving past decimal points; continue aligning them straight down for accurate results.

Order of operations

One key concept in Intermediate Algebra is the order of operations, often remembered with the acronym PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction. It tells us the sequence to follow for operations in expressions.In the exercise, an important first step was to simplify the expression inside the brackets.

• Initially, the problem had us look at \([0.1 + (-0.6) + 8.1]\) and focus on operations within the brackets first.
• By solving the bracketed expression to simplify it first, we maintain logical order and accuracy throughout calculations.

Understanding the need to work through operations in the correct order allows us to systematically and correctly simplify complex expressions.

Simplifying expressions

Simplifying expressions involves reducing them into simpler forms, making calculations easier to complete. The goal is always to break down complicated problems into manageable pieces. Simplicity is key in ensuring understanding and accuracy.The provided exercise started by simplifying inside the brackets first—key to managing expressions with multiple terms.

• Simplifying \([0.1 + (-0.6) + 8.1]\) was broken down in step-by-step fashion, handling two terms at a time: first 0.1 with -0.6, then the result with 8.1.
• Finally, simplifying the resulting term with any terms outside the brackets gave us a neat, straightforward expression to calculate: \[-3.7 + 7.6 = 3.9\].

Simplification requires careful attention to detail and consistent use of basic arithmetic operations. It's like peeling away layers to reveal the core, simplest form of your mathematical problems.