Problem 66 - Intermediate Algebra

Question

In the following exercises, simplify. $$ 15-|3(8-5)| $$

Step-by-Step Solution

Answer

6

Step 1: Simplify inside the parentheses

First, simplify the expression inside the parentheses: (8 - 5) = 3.

Step 2: Multiply by 3

Next, multiply the result by 3: 3 * 3 = 9.

Step 3: Apply the absolute value

Now, apply the absolute value: |9| = 9.

Step 4: Subtract from 15

Finally, subtract the result from 15: 15 - 9 = 6.

Key Concepts

Order of Operations

To simplify algebraic expressions, it's essential to follow the correct sequence, known as the order of operations. This ensures that we get the correct answer every time.

• First, we handle expressions inside parentheses \( ( ... ) \).
• Then, perform any multiplication or division from left to right.
• Finally, carry out addition and subtraction from left to right.

This sequence is commonly remembered using the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). For our example, we first simplified the part inside the parentheses.
Then, we dealt with multiplication, and finally, we performed the subtraction. By following PEMDAS, we ensure accuracy.

Absolute Value

The absolute value of a number is its distance from zero on the number line, regardless of direction. It's denoted by vertical bars, like this: \(| ... |\).
Absolute values turn negative numbers into positive ones and leave positive numbers unchanged.
For instance, \(|-9| = 9\) and \(|9| = 9\).
In our problem, after handling parentheses and multiplication, we calculated the absolute value of 9. As 9 is already positive, \(|9|\) simply remains 9.

Subtraction

Subtraction is one of the basic arithmetic operations where we find the difference between numbers or expressions.
The general formula is \( a - b \), where a is the minuend (the starting number), and b is the subtrahend (the number to be taken away).
In our exercise, after simplifying the absolute value, we subtracted 9 from 15.
Thus \( 15 - 9 = 6 \). Properly handling subtraction steps ensures the correct result.

Multiplication

Multiplication is another fundamental arithmetic operation used to calculate the product when a number is taken a certain number of times.
The general form is \(\text{number} \times \text{number} = \text{product}\).
In our example, after simplifying inside parentheses, we multiplied 3 by 3, resulting in 9.
Properly handling multiplications in the order of operations sequence is crucial for accuracy.