We test integer values of \( x \) from \( -5 \) to \( 5 \), since \( x^2 \leq 25 \Rightarrow |x| \leq 5 \).

["# Testing Integer Values of ( x ) from -5 to 5: Why ( x^2 \leq 25 ) Defines the Range", "When solving mathematical problems involving squares, especially involving expressions like ( x^2 ), defining the domain of integer values is essential for accuracy and clarity. One key inequality commonly tested is ( x^2 \leq 25 ). This condition directly leads to testing integer values of ( x ) from ( -5 ) to ( 5 ), because it hinges on the absolute value: ( |x| \leq 5 ).", "In this article, we’ll walk through why testing integers from (-5) to (5) is fundamental, explain the mathematical reasoning, and highlight why this range is both rigorous and practical.", "---", "## The Mathematical Basis: ( x^2 \leq 25 ) Implies ( |x| \leq 5 )", "The inequality ( x^2 \leq 25 ) describes all real numbers whose square is 25 or less. Taking square roots gives:", "[\n- \sqrt{25} \leq x \leq \sqrt{25}\n]", "[\n-5 \leq x \leq 5\n]", "Since we are limited to integer values, the integers satisfying this condition are:", "[\nx \in {-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5}\n]", "Thus, testing all integers in this closed interval is logically necessary to fully satisfy the inequality.", "---", "## Why Test Integer Values Specifically?", "Testing integers such as ( x = -5 ) through ( x = 5 ) ensures:\n- Completeness — no possible values are missed.\n- Practicality — only integers are typically required in problems involving discrete steps, inequalities, or computational checks.\n- Consistency — ( x^2 ) is symmetric, so testing both negative and positive integers efficiently confirms all cases.", "---", "## Verifying the Range Through Key Values", "Let’s verify why ( x = -5 ) to ( 5 ) cover all valid solutions:", "| ( x ) | ( x^2 ) | Condition ( x^2 \leq 25 ) |\n|--------|-----------|-----------------------------|\n| -5 | 25 | ( 25 \leq 25 ) → True |\n| -4 | 16 | ( 16 \leq 25 ) → True |\n| -3 | 9 | ( 9 \leq 25 ) → True |\n| -2 | 4 | ( 4 \leq 25 ) → True |\n| -1 | 1 | ( 1 \leq 25 ) → True |\n| 0 | 0 | ( 0 \leq 25 ) → True |\n| 1 | 1 | ( 1 \leq 25 ) → True |\n| 2 | 4 | ( 4 \leq 25 ) → True |\n| 3 | 9 | ( 9 \leq 25 ) → True |\n| 4 | 16 | ( 16 \leq 25 ) → True |\n| 5 | 25 | ( 25 \leq 25 ) → True |", "At ( x = -5 ) and ( x = 5 ), equality holds — these bounds confirm the square reaches its maximum at ( \pm 5 ).", "---", "## Applications and Significance", "Understanding and testing values in this range supports:\n- Algebra verification — confirming identities and inequalities.\n- Programming logic — iterating over bounded domains efficiently.\n- Graphical interpretation — plotting the parabola ( y = x^2 ) over ( [-5, 5] ) and verifying all points lie under or on the line ( y = 25 ).\n- Problem-solving frameworks — foundational for students and engineers approaching quadratic behavior.", "---", "## Conclusion", "Testing integer values of ( x ) from ( -5 ) to ( 5 ) is more than a routine check — it’s the logical basis for validating the inequality ( x^2 \leq 25 ). Grounded in the principle that ( |x| \leq 5 ), this range ensures completeness, accuracy, and clarity. Whether solving algebra problems, verifying graphs, or writing code, recognizing and employing this domain grounds your approach in sound mathematical reasoning.", "If you're studying inequalities, quadratic functions, or integer approximations, always start by testing ( x ) in ( [-5, 5] )—your solution will be both correct and well-founded.", "---", "## Key Search Terms (for SEO)", "- Test integer values of ( x ) from -5 to 5\n- Solve ( x^2 \leq 25 ) with integers\n- Why ( x \in [-5,5] ) for ( x^2 \leq 25 )\n- Step-by-step integer domain for quadratic inequalities\n- Validate inequality with integer boundary testing", "---", "Keywords optimized for educational and mathematical SEO targeting students, teachers, and self-learners seeking clarity on testing integer domains in quadratic inequalities."]