Problem 74 - Beginning and Intermediate Algebra

Question

Perform each indicated operation. \(|5-3(9)|-7(-4)\)

Step-by-Step Solution

Answer

The answer is 50.

Step 1: Simplify the expression inside the absolute value

Identify and simplify the expression inside the absolute value. Here, you have the expression inside the absolute value as: \[ 5 - 3(9) \]Simplify it: \[ 5 - 27 = -22 \]

Step 2: Take absolute value

Now, take the absolute value of the result from Step 1: \[ |-22| = 22 \]

Step 3: Simplify the remaining expression

Substitute the absolute value back into the original expression and simplify it:\[ 22 - 7(-4) \]First, compute the multiplication:\[ -7(-4) = 28 \]

Step 4: Perform the final subtraction

Finally, perform the subtraction:\[ 22 + 28 = 50 \]

Key Concepts

Simplifying Expressions

At the heart of many algebra problems is the ability to simplify expressions effectively. This means breaking down complex expressions into simpler components that are easier to work with. Simplifying can involve:

• Combining like terms
• Using the distributive property
• Performing basic arithmetic operations in the correct order

In our exercise, we first deal with the expression inside the absolute value:
n 5 - 3(9)
Using the order of operations (PEMDAS/BODMAS), we perform the multiplication first:
5 - 27

This simplifies to:
-22

By focusing on breaking down the problem into smaller steps, you can make the process more manageable and less intimidating.

Absolute Value

The absolute value of a number is its distance from zero on the number line, regardless of direction. It is always a non-negative value. Absolute values are denoted by vertical bars, like this: |x|.

For example:

• |-5| = 5
• |17| = 17

In our exercise, after simplifying the inside expression to -22, we now take the absolute value:
|-22| = 22

This step ensures that the result is non-negative, simplifying further calculations. Remember, the absolute value operation can change the problem significantly, so handle it with care.

Multiplication and Subtraction in Algebra

Once you've handled absolute values, you may need to combine additional algebraic operations such as multiplication and subtraction. In the given exercise, after finding the absolute value, the expression becomes:

22 - 7(-4)

It's crucial to remember that subtracting a negative is the same as adding the positive equivalent. First, compute the multiplication:
-7(-4) = 28

Now, substitute this value back into the expression and perform the addition:

22 + 28 = 50

Here we see how proper handling of operations in the correct order leads to the final answer. Understanding these core operations makes solving complex algebra problems much more straightforward.