Problem 12 - Elementary Algebra

Question

Simplify the following problems. $$ -16+(-18) $$

Step-by-Step Solution

Answer

Answer: -34

Step 1: Identify the Signs of the Numbers

We can see that both numbers are negative: $$ -16 + (-18) $$

Step 2: Add the Absolute Values of the Given Numbers

Since both numbers are negative, we can add their absolute values without considering the signs: $$ |{-16}| + |{-18}| = 16 + 18 = 34 $$

Step 3: Apply the Proper Sign for the Sum

As both original numbers were negative, the sum should also be negative. So, the simplified expression is: $$ - (34) = -34 $$ The simplified expression is: $$ -16+(-18) = -34 $$

Key Concepts

Understanding Absolute Values

Absolute values are essential in mathematics because they provide the size or magnitude of a number without regard for its sign. For any given number, its absolute value is always a positive number or zero.
The absolute value of a number is denoted by two vertical lines, for example, \(|x|\). Here's what you need to know:

• The absolute value of a positive number is the number itself.
• The absolute value of a negative number is the positive version of that number.
• The absolute value of zero is zero.

For instance, the absolute value of \(-16\) is \(|-16| = 16\), and for \(-18\), it's \(|-18| = 18\). These concepts are crucial in operations like integer addition, particularly when dealing with negative numbers.

Working with Negative Numbers

Negative numbers are numbers less than zero and are represented with a minus sign \(-\) before them. Understanding how to deal with negative numbers is vital as they frequently appear in various applications, such as in temperature scales or financial calculations.
Here are some basics on operations involving negative numbers:

• When you add two negative numbers together, the result is always negative.
• When you subtract a negative number, it is similar to adding its absolute value.
• Adding a positive number to a negative number involves subtracting their absolute values and keeping the sign of the number with the larger absolute value.

For the problem \(-16 + (-18)\), both numbers are negative. Thus, combining them results in another negative number. The process of adding negative numbers involves adding their absolute values and then applying a negative sign to the result.

Simplifying Expressions Made Easy

Simplifying expressions means breaking down complex expressions into simpler and more manageable forms. This often involves combining like terms or using basic arithmetic operations to simplify the expression step by step.
For an expression like \(-16 + (-18)\), the simplification process involves a few crucial steps:

• Identify the signs: Both terms here are negative.
• Use absolute values: Add \(16 + 18\), which equals \(34\).
• Apply the sign: Since both original terms were negative, the expression retains the negative sign, simplifying to \(-34\).

By mastering these steps, you can simplify similar expressions confidently. It helps ensure that you're managing the complexity of problems systematically, resulting in accurate outcomes.