Problem 84 - College Algebra with Modeling and Visualization

Question

Solve the equation. $$ \frac{2 x}{x-3}=-4 $$

Step-by-Step Solution

Answer

The solution is \(x = 2\).

Step 1: Understand the Equation

The given equation is \( \frac{2x}{x-3} = -4 \). We need to solve for the variable \( x \) by isolating it on one side of the equation.

Step 2: Eliminate the Fraction

Multiply both sides of the equation by \(x - 3\) to eliminate the fraction. This results in \(2x = -4(x-3)\).

Step 3: Distribute and Simplify

Expand the right side by distributing -4. The equation becomes \(2x = -4x + 12\).

Step 4: Isolate Variable Terms

Add \(4x\) to both sides to gather all \(x\) terms on one side of the equation: \(2x + 4x = 12\). This simplifies to \(6x = 12\).

Step 5: Solve for \(x\)

Divide both sides by 6 to solve for \(x\): \(x = \frac{12}{6}\), which simplifies to \(x = 2\).

Step 6: Confirm Solution

Plug \(x = 2\) back into the original equation to check: \(\frac{2(2)}{2-3} = -4\) simplifies to \(-4 = -4\), confirming the solution is correct.

Key Concepts

Understanding Rational Equations

In algebra, rational equations are equations that involve fractions which contain polynomials in their numerators and denominators. The term "rational" relates to the word "ratio," as these equations often express the ratio of two polynomials. When you see equations like \( \frac{2x}{x-3} = -4 \), you're dealing with a rational equation.
because the term \( \frac{2x}{x-3} \) is a rational expression.
When solving rational equations, a key step is to eliminate the fractions to simplify the process of finding the variable. This often involves multiplying through by a common denominator, which clears out the fractions and leaves you with a more straightforward equation to solve. Solving rational equations can sometimes reveal extraneous solutions, which arise from the domain restrictions that come with the denominators. Always check your solutions to ensure they don't make any denominator zero.

Methods for Solving Equations

The goal when solving equations is to find the value of the variable that makes the equation true. For rational equations, as demonstrated in the exercise \( \frac{2x}{x-3} = -4 \), the method involves several steps to isolate the unknown variable.

• **Eliminate Fractions:** Multiply both sides by the denominator to clear out the fractional component.
• **Simplify Equations:** Distribute and combine like terms where necessary.
• **Isolate the Variable:** Move all terms involving the variable to one side of the equation.
• **Solve for the Variable:** Simplify the equation to obtain the variable by itself on one side.

Such systematic approaches help break down even complex algebraic problems into manageable steps, enabling solutions to be reached logically and efficiently. In the example, once fractions were cleared, it became straightforward to reach the solution \( x = 2 \). Check solutions by substituting back into the original equation to ensure accuracy.

Key Principles of Algebraic Manipulation

Algebraic manipulation is a core skill when working with equations, including rational ones. It involves rearranging and simplifying expressions to solve for variables. This practice requires familiarity with algebraic rules and properties to ensure each step brings us closer to finding the solution. Common manipulations include:

• **Distributive Property:** Expanding expressions like \( a(b+c) \) to \( ab + ac \) as needed.
• **Combining Like Terms:** Gathering similar terms to simplify expressions, such as combining \( 2x + 4x \) to get \( 6x \).
• **Inverse Operations:** Using opposite operations, like adding to cancel subtraction or multiplying to offset division, to isolate the variable.

These manipulation skills made it possible to turn the initial fraction into a solvable equation, then adjust the expression to reveal that \( x = 2 \). Finally, retracing the solution steps by plugging back values confirms correctness and understanding, crucial for mastery in algebra.