Solve each quadratic inequality. Give the solution set in interval notation. See Exam-ples 5 and 6. x^2+3x-4<0

Verified step by step guidance

1. First, solve the related quadratic equation \(x^2 + 3x - 4 = 0\) to find the critical points.
2. Factor the quadratic equation, if possible, or use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to find the roots.
3. Once the roots are found, these values divide the number line into intervals. Test a point from each interval in the inequality \(x^2 + 3x - 4 < 0\) to determine where the inequality holds true.
4. Determine which intervals satisfy the inequality by checking if the test points make the inequality true.
5. Express the solution set in interval notation, including only the intervals where the inequality is satisfied.

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. This is essential for solving quadratic inequalities, as it allows one to identify the roots of the equation, which are the points where the expression equals zero. For example, the quadratic x^2 + 3x - 4 can be factored to find its roots, aiding in determining the intervals for the inequality.