Solve each quadratic inequality. Give the solution set in interval notation. 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. Identify 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 intervals: \((-\infty, 2)\), \((2, 5)\), and \((5, \infty)\). Test a value from each interval in the inequality \((x - 5)(x - 2) > 0\) to determine 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 set greater than or less than zero (or another value). Solving it means finding all x-values that make the inequality true, often by 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, which are critical points dividing the number line into intervals for testing the inequality.

Interval Notation and Sign Analysis

Interval notation expresses solution sets as ranges of values. After finding roots, the number line is split into intervals. Testing a point from each interval in the inequality determines where the expression is positive or negative, guiding the correct solution intervals.