Problem 6 - Precalculus Student Solutions Manual 5th

Question

Decide what values of the variable cannot possibly be solutions for each equation. Do not solve. $$\frac{5}{2 x}+\frac{2}{x}=6$$

Step-by-Step Solution

Answer

The variable cannot be 0.

Step 1: Identify Denominators

First, identify the denominators in the equation \(\frac{5}{2x} + \frac{2}{x} = 6\). The denominators are \(2x\) and \(x\).

Step 2: Set Denominators Not Equal to Zero

Next, set each denominator not equal to zero to find the values for which the equation is undefined: \(2x eq 0\) and \(x eq 0\).

Step 3: Solve for Variables

Solve each inequality. For \(2x eq 0\), divide by 2 to get \(x eq 0\). For \(x eq 0\), it is already solved. So, \(x eq 0\).

Step 4: Conclusion

Since \(x eq 0\) for both denominators, the variable cannot be 0. Therefore, \(x = 0\) is not a possible solution for this equation.

Key Concepts

Denominators

Understanding the role of denominators in an equation is essential for determining variable restrictions. A denominator is the bottom part of a fraction. For example, in the equation \(\frac{5}{2x} + \frac{2}{x} = 6\), the denominators are \(2x\) and \(x\). Denominators are crucial because they must not equal zero. When a denominator is zero, the fraction becomes undefined and makes the entire equation invalid. In this equation, you need to ensure that both denominators, \(2x\) and \(x\), are never zero to keep the equation valid and solvable.

Undefined Values

Undefined values occur in an equation when a denominator is equal to zero. For the equation \(\frac{5}{2x} + \frac{2}{x} = 6\), setting denominators to zero will make the fractions undefined. We must set each denominator not equal to zero to avoid this. For \(2x eq 0\) and \(x eq 0\), solving these inequalities means \(x eq 0\). If \x\ were zero, it would make the denominators zero which leads the fraction to be undefined. Thus, ensuring denominators do not equal zero avoids undefined values and supports valid solutions.

Variable Constraints

Variable constraints are the specific values that a variable cannot take in an equation. Constraints arise from the need to maintain valid mathematical operations. In the equation \(\frac{5}{2x} + \frac{2}{x} = 6\), variable constraints are determined by the denominators. Both denominators, \(2x\) and \(x\), must not be zero. By solving \(2x eq 0\) and \(x eq 0\), we find the constraint \(x eq 0\). Therefore, the variable \x\ cannot be zero as it would invalidate the equation. Recognizing this constraint is key to understanding the limits within which the variable can operate.