Solve each quadratic inequality. Give the solution set in interval notation. See Example 1. x² + x - 30 ≤ 0

Verified step by step guidance

1. Start by rewriting the inequality: \(x^2 + x - 30 \leq 0\).
2. Factor the quadratic expression on the left side. Look for two numbers that multiply to \(-30\) and add to \$1\(. This gives the factors: \)(x + 6)(x - 5)$.
3. Rewrite the inequality using the factors: \((x + 6)(x - 5) \leq 0\).
4. Determine the critical points by setting each factor equal to zero: \(x + 6 = 0\) gives \(x = -6\), and \(x - 5 = 0\) gives \(x = 5\). These points divide the number line into three intervals: \((-\infty, -6)\), \((-6, 5)\), and \((5, \infty)\).
5. Test a value from each interval in the inequality \((x + 6)(x - 5) \leq 0\) to determine where the product is less than or equal to zero. Then, write the solution set in interval notation including the points where the expression equals zero.

Key Concepts

Quadratic Inequalities

A quadratic inequality involves a quadratic expression set to be greater than, less than, or equal to a value, such as x² + x - 30 ≤ 0. Solving it means finding all x-values that satisfy the inequality, often by analyzing the sign of the quadratic expression over different intervals.

Factoring Quadratic Expressions

Factoring is the process of rewriting a quadratic expression as a product of two binomials, like x² + x - 30 = (x + 6)(x - 5). This helps identify the roots or zeros of the quadratic, which are critical points for determining where the expression changes sign.

Interval Notation and Sign Analysis

Interval notation expresses solution sets as ranges of values, such as [−6, 5]. After finding roots, sign analysis tests values in intervals between roots to determine where the quadratic is positive or negative, guiding the correct interval for the inequality solution.