Problem 36 - 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] $$\frac{7(-2)-6}{-10}$$

Step-by-Step Solution

Answer

The simplified expression is 2.

Step 1: Simplify the Numerator

Focus on the numerator of the fraction, which is \(7(-2) - 6\). First, perform the multiplication: \(7 \times (-2) = -14\). Then, substitute the result into the expression: \(-14 - 6\).

Step 2: Complete the Numerator Simplification

Continue simplifying the numerator by completing the subtraction: \(-14 - 6 = -20\). Now, the entire expression becomes \(\frac{-20}{-10}\).

Step 3: Simplify the Fraction

The expression \(\frac{-20}{-10}\) can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 10. So, \(-20 \div -10 = 2\), because dividing a negative by a negative results in a positive.

Key Concepts

Simplifying Expressions

Simplifying expressions is like tidying up your math problem. The aim is to make the expression as simple as possible without changing its value. Think of it like cleaning a messy room, where you put things in order to see them more clearly.

One of the core rules while simplifying is using the correct 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 helps in determining which operations to perform first.

• Start by simplifying inside any parentheses or brackets.
• Next, calculate any exponents.
• Then, perform multiplication or division, moving from left to right.
• Finally, complete any addition or subtraction, again working from left to right.

When faced with an expression like the one in our example, identifying and grouping operations such as multiplication or addition can help in breaking the problem down into manageable steps before final simplification.

Fractions

Fractions represent a part of a whole and can sometimes seem complex, but they just require some specific steps to simplify. The key parts of a fraction are the numerator (the top number) and the denominator (the bottom number).

To simplify a fraction:

• Look to divide both the numerator and the denominator by their greatest common divisor (GCD).
• The GCD is the largest number that can evenly divide both numbers.

In the exercise provided, once the numerator is simplified to \(-20\), you have the fraction \(-\frac{20}{10}\). Here, the GCD of 20 and 10 is 10, which simplifies to 2 since both the numerator and denominator are negative, resulting in a positive outcome when divided.

Remember, simplifying fractions not only makes them easier to understand, but it often provides clearer insights into the problem.

Integers

Integers include all whole numbers and their negatives, making them crucial in many math problems. They can be positive, negative, or zero, and understanding their properties helps in operations such as multiplication, division, and more.

There are a few rules to keep in mind:

• Multiplying or dividing two negative integers results in a positive integer.
• If you multiply or divide a positive and a negative integer, the result is a negative integer.

In our problem \(7(-2)\), the multiplication of a positive number by a negative number gives a negative result (\(-14\)). Similarly, \(-20\div -10\) results in a positive 2, since dividing two negatives turns into a positive.

Grasping these basics ensures you can correctly handle operations involving integers in any mathematical expression, simplifying the task at hand.