Problem 58 - Elementary Algebra

Question

For the following problems, reduce each rational expression if possible. If not possible, state the answer in lowest terms. \(\frac{-3 x+10}{10}\)

Step-by-Step Solution

Answer

Answer: The simplified form of the rational expression is \(\frac{3x-10}{-10}\).

Step 1: 1. Identify the common factors

The given expression is \(\frac{-3x+10}{10}\). Let's try to find a common factor for the terms in the expression.

Step 2: 2. Factor the expression

We see that both terms in the numerator have a common factor of \((-1)\), and both terms in the denominator have a common factor of \(10\). So, we can factor the expression as: \(\frac{-3x+10}{10} = \frac{-1(3x-10)}{10}\)

Step 3: 3. Cancel out common terms

Since there are no common factors between the numerator and the denominator except the \((-1)\), we cannot further cancel out any terms, and the answer is already in its lowest terms.

Step 4: 4. Write the simplified answer

Therefore, the simplified expression is: $$\frac{-3x+10}{10} = \frac{-1(3x-10)}{10} = \boxed{\frac{3x-10}{-10}}$$

Key Concepts

Algebraic Expressions

The world of algebra involves a variety of mathematical expressions, one of which is the rational expression. Algebraic expressions are combinations of numbers, variables, and arithmetic operations. When we deal with rational expressions, we are looking at ratios (or fractions) of two algebraic expressions. They take the form \( \frac{P(x)}{Q(x)} \) where \( P(x) \) and \( Q(x) \) are polynomial expressions, and \( Q(x) \) is not equal to zero. Simplifying these expressions can make them easier to work with, especially when solving equations or inequalities.

Factoring

Factoring is a critical skill in algebra that simplifies expressions and solves equations. To factor means to write an expression as the product of its factors, which are expressions that can be multiplied together to get the original expression. For example, when you look at the numerator of \( -3x+10 \) you might see that both -3 and 10 have a -1 that can be factored out, transforming the expression to \( -1(3x-10) \). Factoring helps reduce algebraic expressions to simpler forms and can reveal common factors in the numerator and denominator that may cancel out.

Common Factors

When simplifying rational expressions, identifying common factors between the numerator and the denominator plays a key role. A common factor is an expression that divides both the numerator and denominator without leaving a remainder. In our example, looking at the expression \( \frac{-3x+10}{10} \), \( -1 \) is a common factor that appears when factoring the numerator. Recognizing and factoring out common factors is essential for simplifying the expression to its lowest terms.

Lowest Terms

Expressing a rational expression in the lowest terms means reducing it such that no further common factors exist between the numerator and the denominator apart from 1. This is akin to simplifying a fraction to its simplest form. In the given exercise, once it is established that there are no further common factors to cancel, the expression is already in its lowest terms. Hence, the rational expression \( \frac{-1(3x-10)}{10} \) simplifies to \( \frac{3x-10}{-10} \) which cannot be reduced further. Ensuring an expression is in its lowest terms helps achieve clarity and simplicity, allowing for straightforward interpretation and comparison of values.