Solve each quadratic inequality. Give the solution set in interval notation. x²-7x+10<0

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Start by rewriting the inequality: \(x^2 - 7x + 10 < 0\).

Factor the quadratic expression on the left side: \(x^2 - 7x + 10 = (x - 5)(x - 2)\).

Identify the critical points by setting each factor equal to zero: \(x - 5 = 0\) gives \(x = 5\), and \(x - 2 = 0\) gives \(x = 2\).

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 negative, then write the solution set in interval notation based on these results.

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주요 개념

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Quadratic Inequalities

A quadratic inequality involves a quadratic expression set less than, greater than, or equal to a value. Solving it means finding all x-values that make the inequality true, often by analyzing the sign of the quadratic expression over 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 for testing intervals in inequalities.

Interval Notation and Sign Analysis

Interval notation expresses solution sets as ranges of values. After finding roots, the number line is divided into intervals where the quadratic's sign is tested. The solution includes intervals where the inequality holds true, written using parentheses or brackets.

Solve each quadratic inequality. Give the solution set in interval notation. x2-7x+10<0