Problem 66 - Elementary Algebra

Question

Find the sums for the the following problems. \([(-2)+(-8)]+[(-3)+(-7)]\)

Step-by-Step Solution

Answer

Answer: The sum of the given expression is -20.

Step 1: Add numbers within the brackets

We start by adding the numbers inside each bracket: \((-2) + (-8) = -10\) \((-3) + (-7) = -10\)

Step 2: Add the sums obtained from the previous step

Now that we have the sums of the numbers inside each bracket, we can add them together to find the final answer: \(-10 + (-10) = -20\) The sum of the given expression is \(-20\).

Key Concepts

Understanding Negative Numbers

In mathematics, negative numbers are essential for representing values less than zero. They appear often in calculations involving debts, temperatures below zero, and algebraic equations. The key feature of negative numbers is the negative sign (-) placed in front of them. Negative numbers are often tricky because they behave differently when added, subtracted, multiplied, or divided compared to positive numbers.

For instance:

• Adding two negative numbers results in a more negative number, e.g., \[ (-3) + (-5) = -8 \].
• Subtracting a negative number is like adding a positive, e.g., \[ 10 - (-2) = 10 + 2 = 12 \].

When working with negative numbers, practice and familiarity are important to make sense of these rules. By providing clear guidance and controlled practice, learners can develop a strong intuitive grasp of these concepts.

Breaking Down Algebraic Expressions

Algebraic expressions consist of numbers, variables, and operations (like addition, subtraction, etc.). Understanding these expressions is crucial for solving complex equations. An algebraic expression might look like \[ x + 5 \] or \[ 3a - 2b + 7 \].

The expression given in our exercise \([(-2)+(-8)]+[(-3)+(-7)]\) is an example of a purely numerical algebraic expression, with no variables, but it still follows algebraic rules.

Some things to keep in mind:

• Expressions can sometimes appear complex, but breaking them into parts (known as terms) can simplify understanding.
• Each term of the expression can be separately calculated by applying the proper operations.
• Practicing with various expressions helps in identifying patterns and strategies for simplifying them.

Understanding how each component interacts within the expression is vital for finding solutions effectively.

Mastering Bracket Calculations

In mathematics, brackets are used to signal that operations inside them should be carried out first, according to the order of operations (often remembered by the acronym PEMDAS—Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This ensures clarity and avoids ambiguity in calculations.

In the exercise \([(-2)+(-8)]+[(-3)+(-7)]\), the brackets indicate specific operations must be done first. By solving the expressions within each bracket separately, we simplify the equation:

• Calculate \[ (-2) + (-8) = -10 \]
• Calculate \[ (-3) + (-7) = -10 \]

Once the operations inside the brackets are conducted, the results can be combined to provide the final answer.

Knowing when and how to correctly work with bracketed expressions is a key skill in problem-solving across all areas of mathematics. It prevents errors and fosters a deeper understanding of mathematical operations.