2(5x - 4) = 7x + 2 → 10x - 8 = 7x + 2 → 3x = 10 → x = 10/3 → not integer.
["Understanding the Solved Equation: 2(5x − 4) = 7x + 2 Explained Step by Step (x Not Integer)", "Solving equations like ( 2(5x - 4) = 7x + 2 ) is a fundamental skill in algebra, often encountered in middle and high school math, as well as early college courses. This particular problem not only tests basic arithmetic and algebraic manipulation but also introduces students to the concept of non-integer solutions — an important distinction when interpreting results.", "In this detailed explanation, we’ll walk through the step-by-step solution of the equation:", "[\n2(5x - 4) = 7x + 2\n]\n→\nStep 1: Expand the left-hand side.\nDistribute the 2 across the parentheses:", "[\n2 \cdot 5x - 2 \cdot 4 = 10x - 8\n]", "Now the equation becomes:", "[\n10x - 8 = 7x + 2\n]", "Step 2: Eliminate like terms to isolate ( x ).\nSubtract ( 7x ) from both sides:", "[\n10x - 7x - 8 = 2 \quad \Rightarrow \quad 3x - 8 = 7x + 2\n]", "Step 3: Bring all ( x )-terms to one side and constants to the other.\nAdd 8 to both sides:", "[\n3x = 7x + 2 + 8 \quad \Rightarrow \quad 3x = 7x + 10\n]", "Now subtract ( 7x ) from both sides:", "[\n3x - 7x = 10 \quad \Rightarrow \quad -4x = 10\n]", "Step 4: Solve for ( x ).\nDivide both sides by (-4):", "[\nx = \frac{10}{-4} = -\frac{5}{2}\n]", "Alternatively, simplifying:", "[\nx = -\frac{10}{4} = -\frac{5}{2}\n]", "Thus, the exact solution is ( x = -\frac{5}{2} ), which is not an integer — a key insight since many initial learners may expect integer answers.", "---", "### Why the Solution Is Not an Integer", "At first glance, algebra may seem tied only to whole numbers, but this problem illustrates how equations can have fractional or even irrational solutions. Here, the coefficient relationships and constant terms were chosen so that ( x ) represents a rational but non-integer fraction.", "Even though ( x = -\frac{5}{2} ) is exact, its decimal form ( -2.5 ) confirms it is not a whole number — a common feature in linear equations involving real coefficients and constant terms.", "---", "### Real-World Relevance", "Understanding equations with non-integer solutions prepares learners for real-world modeling where outcomes aren’t always whole numbers — for example:", "- Financial calculations involving partial currency units (e.g., currency exchange in fractional cents in some systems)\n- Scientific measurements where precision requires decimal accuracy\n- Engineering problems with tolerances or continuous variables", "Recognizing that solutions can be fractions or decimals helps develop intuition for interpreting data and avoiding assumption errors.", "---", "### Final Thoughts", "The equation ( 2(5x - 4) = 7x + 2 ), while seemingly simple, opens a deeper discussion about solution types beyond integers. Always methodically distribute, combine like terms, isolate variables, and confirm your answer by back-substitution to ensure accuracy.", "Always remember:\nSolving equations doesn’t just find numbers — it builds logical reasoning and precision essential for advanced math and practical problem-solving.", "---", "Summary – Key Takeaways:\n- Step-by-step algebraic manipulation ensures correctness\n- Coefficients and constants influence solution type (integer or fraction/decimal)\n- Non-integer solutions are valid and common in algebra\n- Practice builds confidence and clarity in math reasoning", "If you’re preparing for exams or learning algebra basics, mastering such equations ensures a strong conceptual foundation — no integer necessary!", "---", "Need more practice? Try solving similar equations to strengthen your algebra skills and confidence!"]
