Solve for x: \( \frac{2x + 5}{3} = \frac{x - 1}{4} + 2 \)

["Solve for x: ( \frac{2x + 5}{3} = \frac{x - 1}{4} + 2 )", "Solving equations involving fractions can feel challenging, but with a clear step-by-step approach, you can find the value of ( x ) with confidence. In this guide, we’ll walk through how to solve the equation:", "[\n\frac{2x + 5}{3} = \frac{x - 1}{4} + 2\n]", "---", "### Step 1: Eliminate the Denominators", "To eliminate the fractions, find the least common denominator (LCD) of 3 and 4, which is 12. Multiply every term by 12:", "[\n12 \cdot \frac{2x + 5}{3} = 12 \cdot \left( \frac{x - 1}{4} + 2 \right)\n]", "Distribute the 12:", "[\n4(2x + 5) = 3(x - 1) + 24\n]", "---", "### Step 2: Expand Both Sides", "Multiply out each term:", "[\n4(2x + 5) = 8x + 20\n]\n[\n3(x - 1) + 24 = 3x - 3 + 24 = 3x + 21\n]", "Now the equation becomes:", "[\n8x + 20 = 3x + 21\n]", "---", "### Step 3: Isolate the Variable", "Subtract (3x) from both sides:", "[\n8x - 3x + 20 = 21\n]\n[\n5x + 20 = 21\n]", "Next, subtract 20 from both sides:", "[\n5x = 1\n]", "---", "### Step 4: Solve for x", "Divide both sides by 5:", "[\nx = \frac{1}{5}\n]", "---", "### Final Answer", "[\n\boxed{x = \frac{1}{5}}\n]", "---", "### Why This Equation Matters", "Understanding how to solve linear equations with fractions strengthens algebraic reasoning and prepares you for more complex algebra, such as solving systems of equations or working with rational expressions.", "If you want to practice more, try similar problems like:", "[\n\frac{3x - 2}{5} = \frac{x + 4}{6} - 1\n]", "Use the same method—clear denominators, simplify, isolate x, and verify your solution.", "---", "Keywords: how to solve ( \frac{2x + 5}{3} = \frac{x - 1}{4} + 2 ), step-by-step algebra, solving linear equations with fractions, algebraic problem solving, solving for x algebraically, common denominators, linear equation solving."]