Solve each inequality. Give the solution set in interval notation. (2x-1)(x+5)<0

Verified step by step guidance

1. Identify the critical points by setting each factor equal to zero: solve \$2x - 1 = 0\( and \)x + 5 = 0\( to find the values of \)x$ where the expression changes sign.
2. Solve \$2x - 1 = 0\( to get \)x = \frac{1}{2}\(, and solve \)x + 5 = 0\( to get \)x = -5\(. These points divide the number line into three intervals: \)(-\infty, -5)\(, \)(-5, \frac{1}{2})\(, and \)(\frac{1}{2}, \infty)$.
3. Choose a test point from each interval and substitute it into the inequality \((2x - 1)(x + 5) < 0\) to determine if the product is positive or negative in that interval.
4. Based on the sign of the product in each interval, identify which intervals satisfy the inequality \((2x - 1)(x + 5) < 0\) (where the product is negative).
5. Write the solution set as the union of intervals where the inequality holds true, using interval notation.

Key Concepts

Factoring and Zero-Product Property

Understanding how to factor expressions and identify zeros is essential. The inequality involves a product of two linear factors, so finding where each factor equals zero helps determine critical points that divide the number line into intervals for testing.

Sign Analysis on Intervals

After finding critical points, the number line is split into intervals. Testing the sign of the product in each interval determines where the inequality holds true. This method helps identify which intervals satisfy the inequality (less than zero).

Interval Notation

Interval notation is a concise way to express solution sets of inequalities. It uses parentheses and brackets to indicate open or closed intervals, representing all numbers between critical points that satisfy the inequality.