Problem 66 - Algebra 1

Question

Evaluate the expression. Use estimation to check your answer. $$ 8.9-(-2.1)-7.3 $$

Step-by-Step Solution

Answer

The solution to the given expression is \(3.7\). Using estimation also confirms that this answer is approximately correct.

Step 1: Apply negative number rules

The operators immediately before and after the parentheses are a subtraction and a negation respectively. According to the rule of signs, subtracting a negative number is the same as adding a positive number. Therefore, the expression can be rewritten as: \(8.9 + 2.1 - 7.3\)

Step 2: Perform addition

First, add \(8.9\) and \(2.1\) to get \(11.0\). The expression can now be rewritten as: \(11.0 - 7.3\)

Step 3: Finish with subtraction

Finally, subtract \(7.3\) from \(11\). That leads to the final answer: \(3.7\)

Step 4: Use estimation to check your work

For estimation purposes, rewrite the original expression, rounding each number to the nearest whole number: \(9 - (-2) - 7\). This simplifies to \(9 + 2 - 7 = 4\). This isn’t exactly \(3.7\), but it’s close, supporting the correctness of the exact calculation.

Key Concepts

Negative Numbers

Understanding negative numbers is crucial for evaluating expressions that involve them. A negative number is any number less than zero, typically represented with a minus sign (e.g., -2, -3.7). When dealing with negative numbers, one of the most important rules is how they interact with operations like subtraction.

For instance, subtracting a negative number is equivalent to adding its positive counterpart. This is due to the double negative rule: when you take away a "negative" quantity, it's the same as adding the opposite of that quantity. So in our exercise, when you see \(8.9 - (-2.1)\), it becomes \(8.9 + 2.1\). This insight simplifies many calculations and helps prevent mistakes during evaluation.

Key takeaways:

• Negative numbers are less than zero.
• Subtracting a negative is like adding a positive.
• Understanding this helps in simplifying expressions efficiently.

Addition and Subtraction

Addition and subtraction are basic operations that work in tandem to solve expressions. Start with addition, which means combining quantities or numbers to get a total. Subtraction, meanwhile, involves taking a portion away from a quantity.

When handling an expression such as \(8.9 + 2.1 - 7.3\), it's essential to perform operations step by step. Start by adding \(8.9 + 2.1 = 11.0\). Next, proceed with the subtraction part: \(11.0 - 7.3\). The order of operations is crucial here, as it ensures that every step of the calculation is accurate and leads to the correct final result.

Important tips:

• Always follow the steps: perform additions before subtractions where applicable.
• Keep the results of each step straightforward and clear to avoid errors.

Estimation Techniques

Estimation techniques can be a powerful way to check the accuracy of your calculations. Estimation involves simplifying numbers to their nearest whole values which helps to give a quick sense of whether your more detailed calculation seems reasonable.

For instance, in the expression \(8.9-(-2.1)-7.3\), rounding each number to the nearest whole number gives us \(9 - (-2) - 7\), simplifying further to \(9 + 2 - 7 = 4\). While the estimated answer of 4 isn't exactly the calculated answer of 3.7, it's close enough to suggest that the original calculation was done correctly.

Benefits of Estimation:

• Serves as a quick check for accuracy.
• Helps simplify complex calculations.
• Useful in problem-solving when precision isn't the main concern.