Problem 37 - 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] $$\frac{2+3(-6)}{4-12}$$

Step-by-Step Solution

Answer

The simplified expression is 2.

Step 1: Simplify the numerator

Start by simplifying the expression in the numerator of the fraction. The numerator is given as \(2 + 3(-6)\). First, handle the multiplication: \(3 \times (-6) = -18\). Then perform the addition: \(2 + (-18) = -16\). So, the numerator simplifies to \(-16\).

Step 2: Simplify the denominator

Now, simplify the expression in the denominator of the fraction. The denominator is \(4 - 12\). Subtract 12 from 4: \(4 - 12 = -8\). Thus, the denominator simplifies to \(-8\).

Step 3: Divide the simplified numerator by the simplified denominator

Divide the simplified numerator by the simplified denominator. We have the expression \(\frac{-16}{-8}\). Dividing \(-16\) by \(-8\) gives \(2\), because a negative divided by a negative is a positive.

Key Concepts

Simplifying Expressions

When you're faced with an expression to simplify, like \(\frac{2+3(-6)}{4-12}\), it's crucial to tackle one operation at a time. This process involves strategically breaking down each part to make it more manageable.

• Start with the numerator and the denominator separately. This ensures clarity as you handle each part of the fraction independently.
• Use the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)). This rule helps you prioritize operations correctly.
• In our example, simplification begins by performing the multiplication in the numerator: \(3 \times (-6)\), which results in \(-18\). Then, add \(2 + (-18)\) to obtain \(-16\).
• For the denominator, simply subtract \(12\) from \(4\), resulting in \(-8\).

Finally, combine these simplified parts. Divide the simplified numerator \(-16\) by the simplified denominator \(-8\), which results in the simplified expression of \(2\). Remember, each step you take should get you closer to the simplest form of the expression.

Fractions

Simplifying fractions involves breaking down both the numerator and the denominator to their basic forms. This means addressing any operations present in each part separately before attempting division:

• With fractions like \(\frac{-16}{-8}\), the goal is to divide their numbers perfectly if possible.
• Notice how both the numerator (-16) and the denominator (-8) are negative. When both are negative, and you divide one by the other, the result is positive. This is because dividing two negative numbers together results in a positive outcome.
• If you have operations in the fraction, make sure to resolve them first, as was done with \(2 + 3(-6)\) and \(4 - 12\) in the problem.

In the end, simplifying fractions often involves canceling out the negatives. For \(\frac{-16}{-8}\), you find that it simplifies neatly to \(2\). Always look out for common factors to simplify fractions further, when applicable.

Negative Numbers

Working with negative numbers can be a bit tricky, but with practice, you will become more confident. Here are some key points to consider when handling negative numbers in expressions:

• When multiplying a positive number by a negative number, like \(3 \times (-6)\), the result is always negative. This transforms to \(-18\) in our problem.
• Adding a negative number is equivalent to subtracting its positive counterpart. So in the expression \(2 + (-18)\), you are essentially computing \(2 - 18\).
• When dividing two negative numbers, such as \(-16 \div -8\), the negatives cancel each other out, resulting in a positive number. This is why \(\frac{-16}{-8}\) gives a positive \(2\).

Understanding these principles allows you to handle negative numbers confidently. It's important to keep these rules in mind as they are fundamental when simplifying and solving expressions involving negatives.