Solve each quadratic inequality. Give the solution set in interval notation. See Exam-ples 5 and 6. x²-7x+10>0

Verified step by step guidance

1. Start by rewriting the inequality: \(x^2 - 7x + 10 > 0\).
2. Factor the quadratic expression on the left side: \(x^2 - 7x + 10 = (x - 5)(x - 2)\).
3. Determine the critical points by setting each factor equal to zero: \(x - 5 = 0\) gives \(x = 5\), and \(x - 2 = 0\) gives \(x = 2\).
4. Use the critical points to divide the number line into three intervals: \((-\infty, 2)\), \((2, 5)\), and \((5, \infty)\). Test a value from each interval in the inequality \((x - 5)(x - 2) > 0\) to see where the product is positive.
5. Based on the test results, write the solution set in interval notation, including only the intervals where the inequality holds true.

Key Concepts

Quadratic Inequalities

A quadratic inequality involves a quadratic expression compared to zero using inequality symbols (>, <, ≥, ≤). Solving it means finding all x-values that make the inequality true. This often requires analyzing the sign of the quadratic expression over different intervals.

Factoring Quadratic Expressions

Factoring rewrites a quadratic expression as a product of two binomials. For example, x² - 7x + 10 factors to (x - 5)(x - 2). Factoring helps identify the roots or zeros of the quadratic, which are critical points for testing intervals in inequalities.

Interval Notation and Sign Analysis

Interval notation expresses solution sets as ranges of values, such as (−∞, 2) ∪ (5, ∞). After finding roots, sign analysis tests the quadratic’s positivity or negativity in intervals between and beyond the roots to determine where the inequality holds.