Solve each quadratic inequality. Give the solution set in interval notation. See Exam-ples 5 and 6. x^2-7x+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. Find two numbers that multiply to \$10\( and add to \)-7\(. This gives: \)(x - 5)(x - 2) < 0$.
3. Identify the critical points by setting each factor equal to zero: \(x - 5 = 0\) gives \(x = 5\), and \(x - 2 = 0\) gives \(x = 2\). These points divide the number line into three intervals: \((-\infty, 2)\), \((2, 5)\), and \((5, \infty)\).
4. Test a value from each interval in the inequality \((x - 5)(x - 2) < 0\) to determine where the product is negative. For example, test \(x=1\) in \((-\infty, 2)\), \(x=3\) in \((2, 5)\), and \(x=6\) in \((5, \infty)\).
5. Based on the test results, write the solution set in interval notation where the inequality holds true (where the product is less than zero).

Key Concepts

Quadratic Inequalities

A quadratic inequality involves a quadratic expression set to be greater than or less than zero (or another value). Solving it means finding the range of x-values that make the inequality true, often by analyzing the sign of the quadratic expression over different intervals.

Factoring Quadratic Expressions

Factoring is rewriting 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 determining where the expression changes sign.

Interval Notation and Sign Analysis

Interval notation expresses solution sets as intervals on the number line. After finding roots, sign analysis tests values in intervals between roots to determine where the inequality holds. This method helps write the solution set accurately in interval notation.