Problem 28 - Algebra and Trigonometry with Analytic Geometry

Question

Solve the equation. $$\frac{6}{2 x+11}+5=5$$

Step-by-Step Solution

Answer

No solution exists for this equation.

Step 1: Isolate the Fraction

We start by moving all other terms to one side of the equation. The equation is \( \frac{6}{2x + 11} + 5 = 5 \). Subtract 5 from both sides to isolate the fraction: \( \frac{6}{2x + 11} = 0 \).

Step 2: Clear the Fraction

Set the numerator equal to zero since a fraction is zero if the numerator is zero (assuming the denominator is not zero). If \( \frac{a}{b} = 0 \), then \( a = 0 \). Therefore, \( 6 = 0 \), but this equation is false, implying there is no solution.

Key Concepts

Understanding Algebra in Rational Equations

Algebra provides the foundation for understanding and solving rational equations. In a rational equation, the main feature is the presence of fractions where the numerator and/or the denominator contains a variable. This type of equation is essential because it helps us comprehend how variables interact within expressions. We frequently encounter rational equations in various scenarios, such as proportions and real-world applications.
When working with rational equations, it is vital to keep track of the variables and constants. Each component plays a critical role in maintaining balance on both sides of the equal sign, much like a seesaw. A change on one side affects the entire equation. Therefore, understanding the algebraic properties that govern these equations allows us to manipulate and simplify them, making it easier to express the relationships they represent. We can then solve these equations to find the values that satisfy them.

The Process of Solving Rational Equations

Solving rational equations involves a series of steps aimed at simplifying and finding values for the variable. The method often begins with isolating the fraction, as seen in the original exercise. This involves removing any constants on one side to purely deal with the fraction or fractions involved. By isolating the key variable-containing term, one can focus solely on resolving it.

• Step 1: Move constants to one side of the equation to isolate the fraction.
• Step 2: Clear the fraction, usually by multiplying both sides of the equation by the denominator to eliminate the fraction altogether.
• Step 3: Solve the resulting simpler equation.

However, it's crucial to check the solutions by substituting them back into the original equation to ensure they do not cause any denominators to become zero, which would make the expression undefined. This verification step is often where students discover extraneous solutions, where potential answers do not genuinely satisfy the initial equation.

The Role of Fractions in Equations

Fractions introduce an additional layer of complexity to algebraic equations. They require careful handling to avoid common errors. Primarily, fractions in equations signal the presence of division, greatly affecting how we manipulate the equation. When working with fractions, it's prudent to recall the fundamental property that a fraction equals zero when its numerator is zero, and the denominator is not zero.
This property was applied in the original exercise to determine the non-existence of a solution. The rationale was straightforward: if the equation simplifies to a fraction where the numerator is zero, and yet logically that cannot occur, then obviously the equation has no solution for conventional values. Therefore, understanding how to manage fractions is essential for both solving equations and knowing when they present impossible scenarios.