Problem 71 - Beginning and Intermediate Algebra

Question

Perform each indicated operation. \(3(-5)+|3-10|\)

Step-by-Step Solution

Answer

-8

Step 1: Simplify the Multiplication

First, simplify the multiplication part of the expression. Calculate \[3(-5)\]which results in \[3 \times (-5) = -15\].

Step 2: Evaluate the Absolute Value

Next, evaluate the absolute value part of the expression. Calculate \[|3-10|\].First, find the difference \[3-10 = -7\],and then take the absolute value which is \[|-7| = 7\].

Step 3: Add the Results from Steps 1 and 2

Finally, add the results from the previous steps together:\[-15 + 7\].Compute the sum:\[-15 + 7 = -8\].

Key Concepts

Multiplication in Algebra

Multiplication is a fundamental operation in algebra. When you multiply numbers, the general rule is to combine them into a single product.
In the exercise, we have the term \(3(-5)\). Here, we multiply 3 by -5.
Remember these key points about multiplication involving negative numbers:

• A positive number times a negative number always gives a negative result.
• For example, \[3 \times (-5) = -15\].

So, in our exercise, \(3 \times (-5)\) simplifies directly to -15.
Always follow the correct order of operations: multiplication before addition or subtraction.

Absolute Value

Absolute value measures the distance a number is from zero on the number line, regardless of direction.
It's always a non-negative number.
In mathematical notation, absolute value is represented by vertical bars, like \(|x|\).
In the exercise, we have \(|3-10|\).
First, solve inside the absolute value: \(3-10 = -7\).
Then, find the absolute value of -7: \(|-7| = 7 \).
Remember these key points about absolute value:

• The absolute value of a positive number is the number itself.
• The absolute value of a negative number is its positive counterpart.
• Example: \(|-7| = 7\text{ and } |7| = 7\).

Absolute value helps in understanding distances and differences in mathematics.

Combining Integers

Combining integers involves adding or subtracting whole numbers, both positive and negative.
The rules change slightly depending on the signs of the integers:
In the exercise, we need to combine the results of the multiplication and absolute value: \( -15 + 7 \).
Here are the general rules:

• When adding integers with different signs, subtract the smaller absolute value from the larger absolute value, and keep the sign of the larger absolute value.
• For our example, \[ -15 + 7\]:
  • Absolute values: 15 and 7.
  • Subtract them: 15 - 7 = 8.
  • Since 15 is larger and negative, our result is -8.

So, \[ -15 + 7 = -8.\]
By following these simple rules, you can easily combine integers in different mathematical situations.