Problem 35 - Prealgebra

Question

Use any of the rules developed in this chapter and the rule for order of operations to simplify each of the following expressions as much as possible. [Examples 6–9] $$\frac{2(-3)+10}{-4}$$

Step-by-Step Solution

Answer

The simplified expression is -1.

Step 1: Evaluate the Numerator

First, evaluate the expression in the numerator: \(2(-3) + 10\). Start by multiplying 2 and -3 to get -6, then add 10 to -6, yielding 4. So, the new expression is \(\frac{4}{-4}\).

Step 2: Simplify the Fraction

Simplify the fraction \(\frac{4}{-4}\). Dividing 4 by -4 gives -1. Therefore, the simplified result is -1.

Key Concepts

Simplifying Expressions

Simplifying expressions means making them more compact and easier to work with without changing their value. This is an essential skill in math as it helps in understanding and solving equations more efficiently.
One common method of simplifying includes combining like terms or using mathematical operations to reduce an expression.

• Identify and perform basic arithmetic operations like addition and subtraction on numbers or terms.


• Apply rules for multiplication and division early in the simplification process to simplify computations.


• Use parentheses and the order of operations (PEMDAS/BODMAS: Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction) to guide the simplification process.

In the given exercise, simplifying the numerator and then the entire fraction led us to the simple result of -1.
Each step followed here is a way of simplification which condenses the expression from its more complex form.

Numerator and Denominator

In a fraction, the numerator is the top part, and the denominator is the bottom part. They are critical figures in any fraction and help determine the fraction's value when divided.
Let's look at their primary roles:

• The numerator represents the number of equal parts being considered, and it's placed above the fraction line.


• The denominator, found below the fraction line, indicates the total number of those equal parts that make up a whole.


• Evaluating the numerator and denominator separately is key to simplifying fractions and expressions that include them.

In our example, by focusing on the numerator first (simplifying it to 4 before dividing by the denominator -4), we created a pathway to understand the fraction better.
This clear distinction and simplification allowed the calculation of the fraction as a whole more straightforwardly, ultimately achieving the end result of -1.

Multiplication and Division

Multiplication and Division are fundamental arithmetic operations that often go hand-in-hand, especially when simplifying expressions.
Here's how they interact in simplifying tasks:

• Multiplication is used to combine numbers, while division is used to split a single number into equal parts.


• When both operations appear in an expression, they are performed from left to right as they appear.


• It's critical to respect the signs of numbers, as multiplying or dividing by negative numbers can change the sign of the result.

In the exercise, we first multiplied 2 by -3 which resulted in -6, respecting multiplication's hierarchical position before addition.
Later, once the fraction \( rac{4}{-4} \) was formed, division was used to simplify the expression to -1.
This highlights the importance of following multiplication and division rules to maintain expression accuracy.