Problem 42 - Beginning and Intermediate Algebra

Question

Find each sum. $$ -3+[3+(-8)] $$

Step-by-Step Solution

Answer

-8

Step 1: Simplify the inner expression

First, simplify the inner expression inside the brackets: \(3 + (-8)\). To do this, add the two numbers together considering their signs.\(3 + (-8) = 3 - 8 = -5\)

Step 2: Substitute the simplified value

Now substitute the simplified value back into the original expression: \(-3 + [-5]\).

Step 3: Simplify the final expression

Finally, add the two numbers together: \(-3 + (-5) = -3 - 5 = -8\).

Key Concepts

Simplify Expressions

When working with mathematical expressions, it's helpful to simplify them step by step. Simplifying the inner expressions first makes solving the overall problem easier.
In this case, we start with the expression \(-3 + [3 + (-8)] \).
Focus first on the brackets. Perform the operations inside the brackets before handling the rest.
For example: \[3 + (-8) = 3 - 8 = -5\].
After simplifying inside the brackets, replace it in the original expression, transforming it into \(-3 + [-5]\).
By handling operations in a structured manner, it's easier to avoid mistakes and reach the correct solution efficiently.

Integer Operations

Integer operations involve adding, subtracting, multiplying, and dividing whole numbers. Addition and subtraction of integers often confuse students because of sign rules.
Here's what you need to remember when dealing with integers:

• When adding two positive integers, the sum is positive (Example: 3 + 5 = 8).
• When adding two negative integers, the sum is negative (Example: -3 + (-5) = -8).
• When adding a positive integer and a negative integer, subtract the smaller absolute value from the larger one and take the sign of the number with the greater absolute value (Example: 3 + (-8) = -5).

In our initial problem, \(3 + (-8)\) represents adding a positive and a negative. Thus, we subtract the smaller absolute value (3) from the larger absolute value (8), and take the sign of the number with the greater absolute value (which is -8), resulting in \(-5\).

Addition of Negative Numbers

Adding negative numbers can be tricky but simple if you understand the rules.
Consolidating the process:

• When both numbers are negative, add their absolute values and prepend the negative sign.
• For example, \(-3 + (-5)\) equals \(-(3 + 5) = -8 \).

Applying this to our problem, the simplified expression became \(-3 + [-5]\).
Since both numbers being added are negative, the final step is straightforward.
Add the absolute values of -3 and -5 to get 8, then place a negative sign: \(-8\).
The solution is \(-8\).
Remember these shortcuts to easily perform addition of negative numbers.