Solve each quadratic inequality. Give the solution set in interval notation. See Example 1. x^2 - 2 > x

Verified step by step guidance

1. Rewrite the inequality in standard form: $x^2-x-2>0$.
2. Factor the quadratic expression: $(x-2)(x+1)>0$.
3. Identify the critical points by setting each factor equal to zero: $x-2=0$ and $x+1=0$, giving $x=2$ and $x=-1$.
4. Use the critical points to divide the number line into intervals: $(-\mathrm{\infty },-1)$, $(-1,2)$, and $(2,\mathrm{\infty })$.
5. Test a point from each interval in the inequality $(x-2)(x+1)>0$ to determine where the inequality holds true.

Key Concepts

Quadratic Inequalities

Quadratic inequalities are expressions that involve a quadratic polynomial set in relation to a value, typically using symbols like '>', '<', '≥', or '≤'. To solve these inequalities, one must find the values of the variable that satisfy the inequality, often by determining the roots of the corresponding quadratic equation and testing intervals between these roots.

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, the interval (2, 5] includes all numbers greater than 2 and up to 5, including 5 but not 2.

Testing Intervals

Testing intervals is a method used to determine where a quadratic inequality holds true. After finding the roots of the corresponding quadratic equation, the number line is divided into intervals based on these roots. By selecting test points from each interval and substituting them back into the inequality, one can identify which intervals satisfy the inequality.