Problem 56 - Elementary Algebra

Question

For the following exercises, perform the indicated operations. $$ [5+(-6)]-[2+(-4)] $$

Step-by-Step Solution

Answer

Question: Evaluate the expression [5+(-6)] - [2+(-4)]. Answer: 1

Step 1: Evaluate the expressions within parentheses

First, we will evaluate the expression within each pair of brackets: $$ [5+(-6)] = 5 - 6 = -1 $$ $$ [2+(-4)] = 2 - 4 = -2 $$ Now we can rewrite the expression with these values: $$ [-1] - [-2] $$

Step 2: Subtract the integers

Now, we need to subtract the values we obtained in Step 1. Recalling that subtraction means the addition of an opposite number, we can write: $$ -1 - (-2) = -1 + 2 $$

Step 3: Add the integers

Finally, we can perform the addition of the integers: $$ -1 + 2 = 1 $$ So, the result of the given expression is 1.

Key Concepts

Addition and Subtraction of Integers

Adding and subtracting integers involve understanding how positive and negative numbers interact. An integer is simply a whole number which can be positive, negative, or zero. When you add two integers, you either combine positive values to get a larger positive number or negative values to get a larger negative number. On the other hand, when subtracting, things can get a bit tricky.
Subtraction can often be seen as adding the opposite. For example, subtracting a negative is the same as adding a positive. This concept can feel puzzling at first, but with practice, it becomes more intuitive.

• When subtracting: \(6 - 3\) is straightforward, yielding 3.
• For \(-6 - 3\): Think of it as \(-6 + (-3) = -9\).
• For \(-6 - (-3)\): Convert to \(-6 + 3 = -3\).

Consistently applying these ideas helps in solving more complex algebraic expressions.

Parentheses Evaluation

Parentheses are often used in mathematics to indicate which operations should be performed first. In the exercise \([5 + (-6)] - [2 + (-4)]\), the parentheses group numbers and operations to prioritize.
The order of operations dictates that expressions within parentheses are evaluated first. Treat each set like its own mini problem, solving them before addressing anything outside.

• For \([5 + (-6)]\), calculate it as \(5 - 6 = -1\).
• Similarly, \([2 + (-4)]\) translates to \(2 - 4 = -2\).

This initial evaluation simplifies the problem, paving the way to focus on the remaining operations with clarity.

Negative Numbers

Negative numbers extend the number line to the left of zero. They can be counterintuitive since they combine with positive numbers in patterns that differ from regular counting.
Understanding their behavior:

• Adding a negative number decreases the total value. For instance, \(5 + (-6)\) results in \(-1\).
• A negative subtracted from a positive reduces the value further. E.g., \(2 - 4 = -2\).
• Subtraction of negatives can be seen as adding their positive counterparts, e.g., \(-1 - (-2)\) becomes \(-1 + 2 = 1\).

This comprehension is essential to solve real-world problems, such as temperature changes or financial calculations.

Operational Order

The order of operations is crucial in solving algebraic expressions, ensuring every operation is performed correctly. Often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)), it guides the sequence of operations.
In our exercise, prioritizing calculations within parentheses is the first step. After evaluating these expressions, subtraction follows, respecting the defined sequence.

• First, resolve operations inside parentheses: \([5 + (-6)]\) and \([2 + (-4)]\).
• Then, address the subtraction of these results: \([-1] - [-2]\).
• Finally, solve by adding the opposite for clarity: \(-1 + 2\).

Mastering this order aids in accurate and efficient mathematical problem-solving.

Algebraic Expressions

Algebraic expressions are mathematical sentences that include numbers, variables, and operational symbols. They require understanding the relationships between different components.
In the given expression \([5+(-6)]-[2+(-4)]\), multiple operations are embedded within each set of parentheses, each interaction defined by its context within the whole.

• The expression inside each bracket is considered individually, as if simplifying separate expressions.
• Applying algebraic principles like 'adding the opposite' transforms the subtraction into a more visible addition \((-1 + 2)\).

This practice builds foundational skills necessary for progressing into more advanced algebraic concepts, enhancing overall mathematical proficiency.