Question: Find the value of $x$ that satisfies $\displaystyle rac{2x - 1}{x + 3} = rac{5}{4}$.

["# How to Solve $\displaystyle \frac{2x - 1}{x + 3} = \frac{5}{4}$: A Step-by-Step Guide", "Solving rational equations is a fundamental skill in algebra, and understanding how to find the value of $x$ that satisfies an equation like $\displaystyle \frac{2x - 1}{x + 3} = \frac{5}{4}$ is essential for mastering more advanced math topics. In this article, we’ll walk through the process step-by-step, explaining how to solve for $x$ safely and clearly—while also offering tips to avoid common mistakes.", "---", "## The Equation to Solve", "We are given:", "$$\n\frac{2x - 1}{x + 3} = \frac{5}{4}\n$$", "This equation involves a rational expression (a fraction with variables), and our goal is to isolate $x$.", "---", "## Step 1: Eliminate the denominators", "To eliminate the fractions, multiply both sides of the equation by the least common denominator (LCD), which in this case is $4(x + 3)$. This step clears the fractions without introducing extraneous solutions—provided we later check that our solution does not make the denominator zero.", "$$\n4(x + 3) \cdot \frac{2x - 1}{x + 3} = 4(x + 3) \cdot \frac{5}{4}\n$$", "Simplify both sides:", "- Left side: $4(x + 3)$ cancels with $(x + 3)$, leaving $4(2x - 1)$\n- Right side: $4$ and $\frac{5}{4}$ cancel partially, leaving $5(x + 3)$", "So we have:", "$$\n4(2x - 1) = 5(x + 3)\n$$", "---", "## Step 2: Expand both sides", "Distribute the constants:", "$$\n8x - 4 = 5x + 15\n$$", "---", "## Step 3: Collect like terms", "Subtract $5x$ from both sides:", "$$\n8x - 5x - 4 = 15\n\Rightarrow 3x - 4 = 15\n$$", "Add 4 to both sides:", "$$\n3x = 19\n$$", "---", "## Step 4: Solve for $x$", "Divide both sides by 3:", "$$\nx = \frac{19}{3}\n$$", "---", "## Step 5: Check for extraneous solutions", "Before accepting the solution, it’s crucial to verify it does not make any denominator zero. Recall the original denominator was $x + 3$.", "Plug in $x = \frac{19}{3}$:", "$$\nx + 3 = \frac{19}{3} + 3 = \frac{19}{3} + \frac{9}{3} = \frac{28}{3} <br/>\neq 0\n$$", "Since the denominator is not zero, the solution is valid.", "---", "## Final Answer", "$$\n\boxed{x = \frac{19}{3}}\n$$", "---", "## Why This Method Works", "- Cross-multiplication with LCD avoids errors from solving one side at a time.\n- Isolating $x$ through standard algebraic operations maintains equality.\n- Domain verification prevents accepting solutions that invalidate the original equation.", "Mastering rational equations builds a strong foundation for algebra and beyond—use these techniques confidently, and tackle more complex problems with ease.", "---", "### Key takeaways:", "- Always eliminate fractions by multiplying both sides by the LCD.\n- Never skip checking that the solution doesn’t make any denominator zero.\n- Practice step-by-step—slow and methodical solutions prevent errors.\n- Use fractions carefully when simplifying expressions.", "Ready to climb your problem-solving ladder? Solving $\displaystyle \frac{2x - 1}{x + 3} = \frac{5}{4}$ is not just about finding $x = \frac{19}{3}$—it’s about developing a powerful algebraic mindset. Start today!"]