Problem 53 - Beginning and Intermediate Algebra

Question

Add See Examples \(\ell\) through 7 . $$ |9+(-12)|+|-16| $$

Step-by-Step Solution

Answer

The result is 19.

Step 1: Evaluate Operations Inside Absolute Values

First, solve inside the absolute value operators. Begin with the expression inside the first absolute value: \[ 9 + (-12) = 9 - 12 = -3 \]Now the expression becomes: \[ |-3| + |-16| \]

Step 2: Apply Absolute Value

Determine the absolute values of each expression:1. For \(-3\), the absolute value is \(|-3| = 3\).2. For \(-16\), the absolute value is \(|-16| = 16\).So the expression now reads: \[ 3 + 16 \]

Step 3: Perform Addition

Add the results from the two absolute values:\[ 3 + 16 = 19 \].

Key Concepts

Integer Addition

Integer addition involves combining whole numbers, which can be positive, negative, or zero. When adding integers, it’s important to pay attention to their signs. Here is a simple rule to follow:

• When the signs are the same, add the numbers and keep the sign. For instance, \(7 + 9 = 16\) because both numbers are positive.
• When the signs are different, subtract the smaller number from the larger number and take the sign of the larger number. In the exercise above, you have \(9 + (-12)\). Here, subtract 9 from 12, giving you 3, but since 12 is larger and negative, the result is \(-3\).

Understanding these rules ensures that you accurately add integers in math problems.

Evaluating Expressions

Evaluating expressions means finding the value of an expression by carrying out all the operations. This often includes several steps like addition, subtraction, multiplication, and division, possibly combined with the use of absolute values or parentheses.

In our given exercise, the expression involves absolute values: \(|9 + (-12)| + |-16|\). To evaluate it correctly, follow these steps:

• First, carry out all operations inside each absolute value symbol. Solve the equation inside, just like you saw with \(9 + (-12)\).
• Apply the absolute value. Remember that absolute value converts any number to a non-negative number. For instance, both \(-3\) and \(3\) will have an absolute value of \(3\).
• Add them together as the final step to find the evaluated value.

Being meticulous with these steps makes it easier to reach an accurate solution.

Order of Operations

The order of operations is an essential set of rules in mathematics dictating the sequence in which operations are performed. This set of rules is often remembered by the acronym PEMDAS:

• P: Parentheses first
• E: Exponents (i.e., powers and square roots, etc.)
• M/D: Multiplication and Division (left-to-right)
• A/S: Addition and Subtraction (left-to-right)

In exercises like the one given, understanding the order ensures operations are executed properly even when absolute values complicate the process.

For instance, in this expression: \(|9 + (-12)| + |-16|\), you should first solve what’s inside the absolute value symbols by following integer addition rules, before you move on to finding the absolute value. After evaluating those parts first, you can finish by carrying out the addition operation. Observing this order helps avoid errors and keeps math fun and logical!