Solve the inequalities in Exercises 35-42 . Express the...

Solve the inequalities in Exercises $35-42 .$ Express the solution sets as intervals or unions of intervals and show them on the real line. Use the result $\sqrt{a^{2}}=|a|$ as appropriate.
$$
x^{2}-x-2 \geq 0
$$

Key Concepts

Interval Notation

Interval notation is a concise way to represent the set of solutions to an inequality. It uses brackets and parentheses to indicate whether endpoints are included in the solution set, which directly corresponds to the nature of the inequality (strict or inclusive).

Critical Points

Critical points are the values of x where the quadratic expression is equal to zero. They act as boundary points between intervals on the number line, and by testing these intervals, one can determine which intervals satisfy the inequality.

Sign Chart Analysis

After factoring and finding the critical points, a sign chart is used to test the sign of the quadratic expression in the intervals determined by those points. This analysis helps in deciding whether the quadratic expression is positive or negative in those intervals.

Factoring Quadratic Expressions

Factoring simplifies the process of solving quadratic inequalities by expressing the quadratic as a product of linear factors. This reveals the roots or zeros of the equation, which are the critical points that divide the real line into intervals for further analysis.

Quadratic Inequalities

A quadratic inequality involves an expression in the form ax² + bx + c compared to 0. Solving these inequalities requires finding where the quadratic expression is nonnegative or nonpositive by identifying its critical points and using them to determine intervals on the number line where the inequality holds.

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