Problem 59 - Elementary and Intermediate Algebra

Question

Simplify. See Example 7 . $$ \frac{2 x-7}{7-2 x} $$

Step-by-Step Solution

Answer

The simplified form of the expression is -1.

Step 1: Identify the Expression

We are given the expression \( \frac{2x - 7}{7 - 2x} \). To simplify this, we first need to understand the structure of the numerator and the denominator.

Step 2: Factor Out Negative One

Notice that both the numerator and the denominator are closely related except for the signs. Observe that \(7 - 2x\) is the negative form of \(2x - 7\). We can rewrite this as \(-1 \times (2x - 7)\). Hence, the expression becomes \(\frac{2x - 7}{-1 \times (2x - 7)}\).

Step 3: Simplify the Expression

After factoring out \(-1\) from the denominator, we can cancel out the \(2x - 7\) term in the numerator with the \(2x - 7\) term in the denominator. This leads to a simplified expression of \(\frac{1}{-1} = -1\).

Key Concepts

Numerator and Denominator

In algebraic expressions, it is crucial to understand the components labeled as the numerator and the denominator. The numerator is the expression located above the fraction line, and it represents the dividend—the part that is being divided. In the exercise given, the numerator is \(2x - 7\). The denominator is below the fraction line, functioning as the divisor. For the given exercise, the denominator is \(7 - 2x\). Recognizing these two parts helps in breaking down expressions for simplification.

• Numerator: In our case, it is \(2x - 7\).
• Denominator: Here, it appears as \(7 - 2x\).

Whenever you face a fraction, identifying these parts helps streamline the process of simplification and manipulation.

Factoring Negative One

Factoring negative one is a simple but powerful technique when simplifying algebraic expressions, especially when dealing with fractions. It involves expressing a term in a form that includes the multiplication of \(-1\). If two expressions seem almost identical but differ in signs, you might be able to use this concept.

• In this exercise, observe that \(7 - 2x\) is \(-1\) times \(2x - 7\).
• By rewriting \(7 - 2x\) as \(-1 \times (2x - 7)\), you've employed factoring negative one effectively.

This approach helps in recognizing opportunities to further simplify expressions by adjusting signs, which often sets up the expression for easy cancellation in the next steps.
Factor by pulling out a negative sign when signs mismatch between terms.

Cancellation Law

The cancellation law is an essential principle in algebra that allows us to simplify fractions by removing common factors from the numerator and the denominator. To apply this law, simply divide both the numerator and the denominator by the expression you're canceling out, as long as it's not zero.

• In the current exercise, after factoring out \(-1\), the expression becomes \(\frac{2x - 7}{-1 \times (2x - 7)}\).
• This setup shows a common factor of \(2x - 7\) in both the numerator and the denominator.

By applying the cancellation law, these terms can be canceled out: \[\frac{2x - 7}{-1 \times (2x - 7)} = \frac{1}{-1} = -1\]Remember, the cancellation law provides a streamlined path to simplify complex expressions and should always be verified for validity—particularly ensuring that the common factor is not equal to zero, as dividing by zero is undefined.