Solve for \( x \) in the inequality \( 3x - 7 < 2x + 5 \).
["Solve for ( x ) in the Inequality ( 3x - 7 < 2x + 5 )", "Understanding how to solve linear inequalities is essential for mastering algebra and developing problem-solving skills. One common type of problem students face is solving inequalities like ( 3x - 7 < 2x + 5 ). In this article, we’ll walk step-by-step through solving this inequality, explain the logic behind each move, and highlight key concepts to help you succeed in math homework, quizzes, or standardized tests.", "---", "### Step 1: Understand the Inequality", "We start with:", "[\n3x - 7 < 2x + 5\n]", "This inequality states that the expression on the left ( ( 3x - 7 ) ) is less than the expression on the right ( ( 2x + 5 ) ). Our goal is to isolate ( x ) and find all values that make this statement true.", "---", "### Step 2: Subtract ( 2x ) from Both Sides", "To collect the ( x )-terms on one side, subtract ( 2x ) from both sides:", "[\n3x - 7 - 2x < 2x + 5 - 2x\n]", "Simplify both sides:", "[\nx - 7 < 5\n]", "This step eliminates ( x ) from the right, making it easier to isolate.", "---", "### Step 3: Add 7 to Both Sides", "Next, add 7 to both sides to isolate ( x ):", "[\nx - 7 + 7 < 5 + 7\n]", "Which simplifies to:", "[\nx < 12\n]", "---", "### Final Answer", "[\n\boxed{x < 12}\n]", "This means that any value of ( x ) less than 12 satisfies the original inequality.", "---", "### Important Notes on Inequalities", "- Direction preservation: When adding or subtracting the same number from both sides of an inequality, the direction (less than or greater than) stays unchanged.\n- Multiplication and division: When multiplying or dividing both sides by a negative number, always reverse the inequality sign. In this problem, we avoided this issue by only adding and subtracting.", "---", "### Why This Inequality Matters", "Linear inequalities are foundational in algebra, used in real-world applications such as budgeting, physics, economics, and engineering. Mastering their solution helps build logical reasoning and analytical thinking.", "---", "### Practice & Next Steps", "To reinforce your skills, try solving similar inequalities:", "- Solve ( 4x - 3 > 2x + 9 )\n- Solve ( 5 - 2x \leq 3x + 1 )", "For further learning, explore systems of inequalities and graphical representations on number lines or coordinate planes.", "---", "TL;DR:\nSolve ( 3x - 7 < 2x + 5 ) by subtracting ( 2x ) from both sides, then isolating ( x ) by adding 7. The solution is ( x < 12 ). Clear, step-by-step algebra makes inequalities manageable—practice often to build confidence!", "---", "Keywords for SEO:\nsolve for ( x ), inequality steps, linear inequality solved, how to solve ( 3x - 7 < 2x + 5 ), step-by-step inequality help, algebra practice, inequality domain, math help for inequalities, algebraic solutions."]
