Use the technique described in Exercises 87–90 to solve each inequality. Write the solution set in interval notation. x^2 - x - 6 < 0

Verified step by step guidance

1. 1. Start by factoring the quadratic expression $x^2-x-6$. Look for two numbers that multiply to -6 and add to -1.
2. 2. Once factored, the expression becomes $(x-3)(x+2)<0$.
3. 3. Identify the critical points by setting each factor equal to zero: $x-3=0$ and $x+2=0$. This gives the critical points $x=3$ and $x=-2$.
4. 4. Use these critical points to divide the number line into intervals: $(-\mathrm{\infty },-2)$, $(-2,3)$, and $(3,\mathrm{\infty })$.
5. 5. Test a point from each interval in the inequality $(x-3)(x+2)<0$ to determine which intervals satisfy the inequality. Write the solution set in interval notation based on these results.

Key Concepts

Inequalities

Inequalities express a relationship where one side is not equal to the other, often using symbols like <, >, ≤, or ≥. In this case, the inequality x^2 - x - 6 < 0 indicates that we are looking for values of x that make the quadratic expression negative. Understanding how to manipulate and solve inequalities is crucial for finding the solution set.

Quadratic Functions

A quadratic function is a polynomial of degree two, typically expressed in the form f(x) = ax^2 + bx + c. The graph of a quadratic function is a parabola, which can open upwards or downwards depending on the sign of 'a'. Analyzing the roots (or zeros) of the quadratic equation helps determine where the function is positive or negative, which is essential for solving the given inequality.

Interval Notation

Interval notation is a mathematical notation used to represent a range of values on the number line. It uses parentheses and brackets to indicate whether endpoints are included (closed interval) or excluded (open interval). For example, the solution set for the inequality will be expressed in interval notation, which succinctly conveys the range of x values that satisfy the inequality.