Problem 93 - Beginning and Intermediate Algebra

Question

Simplify. $$ \frac{22+(3)(-2)}{-5-2} $$

Step-by-Step Solution

Answer

The simplified result is \( -\frac{16}{7} \).

Step 1: Simplify the Numerator

First, let's simplify the expression in the numerator. We have \( 22 + (3)(-2) \). Start by calculating \( (3)(-2) \), which equals \(-6\). Then add \(-6\) to \(22\): \[ 22 + (-6) = 16 \].

Step 2: Simplify the Denominator

Now, simplify the denominator, which is \(-5 - 2\). Add \(-5\) and \(-2\) to get:\[ -5 - 2 = -7 \].

Step 3: Divide Numerator by Denominator

Now, divide the simplified numerator by the simplified denominator: \[ \frac{16}{-7} \]. When dividing a positive number by a negative number, the result is negative, so this fraction is:\[ -\frac{16}{7} \].

Key Concepts

Numerator and Denominator

In a fraction, there are two main parts: the numerator and the denominator. The numerator is the number above the fraction line, and it represents the part we are considering. In our exercise, the original numerator is written as \(22 + (3)(-2)\).

The denominator, on the other hand, is the number below the fraction line. It indicates the total number of equal parts the whole is divided into or, in some cases, a modifier for the numerator’s value. In our case, the original denominator is \(-5 - 2\).

Simplifying either the numerator or the denominator helps make the fraction easier to work with. Once simplified, both parts clearly give a straightforward fraction to interpret. To simplify, you perform basic operations like addition and multiplication on each part separately, ensuring the arithmetic is cleanly performed before any division is done.

Multiply Integers

Multiplying integers is a simple but essential arithmetic operation that is used in algebra simplification. It involves taking two numbers and adding one of them to itself as many times as the other number indicates. However, a quicker approach is to memorize and apply the rules for multiplying numbers with different signs.

For example, consider multiplying \(3\) by \(-2\). When multiplying a positive integer by a negative integer, the result is always negative. Thus \((3)(-2)\) gives us \(-6\).

Here are some rules for multiplying integers:

• Positive × Positive = Positive
• Positive × Negative = Negative
• Negative × Positive = Negative
• Negative × Negative = Positive

Making use of these rules ensures your integer multiplication is accurate, concretely helping with the first step of algebraic simplification in our fraction exercise.

Division of Fractions

In dividing fractions, we take a simplified numerator and denominator from previous computations and express them as a single quotient. Our exercise leads us to a fraction \(\frac{16}{-7}\). To execute the division, the numerator \(16\) is divided by the denominator \(-7\).

A key point in fraction division is that a positive number divided by a negative number yields a negative result. That means our final expression is \(-\frac{16}{7}\).

Division with integers and fractions requires careful attention to signs. Here’s a tip for tackling similar problems:

• Ensure both numerator and denominator are fully simplified before dividing.
• Remember that opposite signs result in a negative quotient.
• Keep the division result in the simplest form unless instructed otherwise.

Even fractions with negative components yield simple results when the rules are consistently applied, illustrating the effectiveness of division in fraction simplification.