Solve each quadratic inequality. Give the solution set in interval notation.. x²+3x-4<0

Verified step by step guidance

1. Start by rewriting the inequality: \(x^2 + 3x - 4 < 0\).
2. Find the roots of the quadratic equation \(x^2 + 3x - 4 = 0\) by factoring or using the quadratic formula. For factoring, look for two numbers that multiply to \(-4\) and add to \$3$.
3. Once factored, express the quadratic as \((x + a)(x + b) = 0\), where \(a\) and \(b\) are the roots found in the previous step.
4. Determine the intervals defined by the roots on the number line. These intervals will be \((-\infty, \text{root}_1)\), \((\text{root}_1, \text{root}_2)\), and \((\text{root}_2, \infty)\).
5. Test a value from each interval in the original inequality \(x^2 + 3x - 4 < 0\) to see where the inequality holds true. The solution set will be the union of intervals where the inequality is satisfied, expressed in interval notation.

Key Concepts

Quadratic Inequalities

A quadratic inequality involves a quadratic expression set less than, greater than, or equal to a value, often zero. Solving it means finding all x-values that make the inequality true. This requires understanding how the parabola defined by the quadratic behaves relative to the x-axis.

Factoring Quadratic Expressions

Factoring rewrites a quadratic expression as a product of two binomials, making it easier to find the roots or zeros. These roots divide the number line into intervals to test for the inequality. For example, x^2 + 3x - 4 factors to (x + 4)(x - 1).

Interval Notation and Test Points

Interval notation expresses solution sets as ranges of values, using parentheses or brackets. After finding roots, the number line is split into intervals. Test points from each interval determine where the inequality holds, allowing the solution to be written in interval notation.