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. 1. Start by identifying the quadratic inequality: $x^2-7x+10<0$.
2. 2. Factor the quadratic expression on the left side: $(x-5)(x-2)<0$.
3. 3. Determine the critical points by setting each factor equal to zero: $x-5=0$ and $x-2=0$, which gives $x=5$ and $x=2$.
4. 4. Use the critical points to divide the number line into intervals: $(-\mathrm{\infty },2)$, $(2,5)$, and $(5,\mathrm{\infty })$.
5. 5. Test a point from each interval in the inequality $(x-5)(x-2)<0$ to determine where the inequality holds true.

Key Concepts

Quadratic Inequalities

Quadratic inequalities are expressions that involve a quadratic polynomial set to be less than, greater than, less than or equal to, or greater than or equal to zero. To solve these inequalities, one typically finds the roots of the corresponding quadratic equation and then tests intervals to determine where the inequality holds true.

Interval Notation

Interval notation is a mathematical notation used to represent a range of values. It uses parentheses and brackets to indicate whether endpoints are included (closed intervals) or excluded (open intervals). For example, (a, b) represents all numbers between a and b, not including a and b, while [a, b] includes both endpoints.

Factoring Quadratics

Factoring quadratics involves rewriting a quadratic expression as a product of its linear factors. For the inequality x^2 - 7x + 10 < 0, factoring helps identify the roots of the equation x^2 - 7x + 10 = 0, which are critical points that divide the number line into intervals for testing the inequality.