Problem 44 - Beginning and Intermediate Algebra

Question

Simplify each expression. (Remember the order of operations.) See Examples 4 and 5 $$ -16-(-3)+(-11)-14 $$

Step-by-Step Solution

Answer

The expression simplifies to -38.

Step 1: Clarify the Expression

First, let's write out the expression clearly. We need to simplify:\[-16 - (-3) + (-11) - 14\]

Step 2: Simplification of Double Negative

The expression contains a double negative operation. Remember that subtracting a negative number is equivalent to adding its absolute value.\[-16 - (-3)\] simplifies to \[-16 + 3\].

Step 3: Perform Additions and Subtractions

Now evaluate from left to right:1. \[-16 + 3 = -13\]2. \[-13 + (-11) = -13 - 11 = -24\]3. \[-24 - 14 = -38\]

Step 4: Result

The simplified value of the expression is:\[-38\].

Key Concepts

Understanding Order of Operations

The order of operations is a fundamental concept in mathematics. It determines the sequence in which the parts of an expression should be evaluated. If not adhered to, you may end up with incorrect results.
Think of it like a recipe: if you don't follow the steps in the right order, your dish might not turn out right. Similarly, in math, you need to know which operations to perform first to simplify correctly.
To remember the sequence, you can use the acronym PEMDAS:

• Parentheses first
• Exponents (i.e., powers and roots, etc.)
• Multiplication and Division (left to right)
• Addition and Subtraction (left to right)

In this exercise, consider subtracting a negative number as a subtraction within parentheses. Simplifying it first is crucial as it closely follows the order of operations. When you encounter expressions with several additions and subtractions, perform them from left to right after dealing with any parentheses and exponents.

Dealing with Double Negatives

Handling double negatives in expressions is like cracking a code. When you see a subtraction followed by a negative number, you're actually dealing with a double negative.
This might seem tricky, but it's not too different from everyday language. For instance, saying "I don't have no money" is another way of saying "I do have money."
In math, subtracting a negative is like taking away a bad thing, which is good. Thus, subtracting a negative number turns into an addition of its absolute value. For example, in \(-16 - (-3)\), subtracting \(-3\) converts to adding 3. It changes the operation from a subtraction to addition, making it \(-16 + 3\). This simplification makes calculations more straightforward.

Addition and Subtraction Simplification

Once you've tackled any double negatives and adjusted for parentheses or complex operations, it's time to focus on addition and subtraction.
These operations are usually straightforward, but remember to move from left to right as you proceed through the expression. This follows the order of operations rule.
Let's break down the steps using the exercise example:

• Start with \(-16 + 3 = -13\).
• Next, handle the addition of a negative, \(-13 + (-11)\), which simplifies to \(-13 - 11 = -24\).
• Finally, complete the final subtraction, \(-24 - 14\), leading to \(-38\).

Keeping these steps clear ensures accuracy and avoids any mix-up or errors in basic arithmetic.