Solve each quadratic inequality. Give the solution set in interval notation. See Example 1. 2^x² + 5 ≤ 11x

Verified step by step guidance

1. Rewrite the inequality to standard quadratic form by moving all terms to one side: \$2x^2 + 5 \leq 11x\( becomes \)2x^2 - 11x + 5 \leq 0$.
2. Identify the quadratic expression: \$2x^2 - 11x + 5\(. To solve the inequality, first find the roots of the corresponding quadratic equation \)2x^2 - 11x + 5 = 0$.
3. Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) with \(a=2\), \(b=-11\), and \(c=5\) to find the roots.
4. Determine the intervals defined by the roots and test a value from each interval in the inequality \$2x^2 - 11x + 5 \leq 0$ to see where the inequality holds true.
5. Express the solution set as an interval or union of intervals based on where the quadratic expression is less than or equal to zero.

Key Concepts

Quadratic Inequalities

A quadratic inequality involves a quadratic expression set less than, greater than, or equal to a value. Solving it requires finding the range of x-values that satisfy the inequality, often by analyzing the related quadratic equation and testing intervals.

Solving Quadratic Equations

To solve a quadratic inequality, first solve the corresponding quadratic equation by setting the expression equal to the boundary value. Techniques include factoring, completing the square, or using the quadratic formula to find critical points.

Interval Notation and Sign Analysis

After finding critical points, use interval notation to express solution sets. Test values in intervals defined by these points to determine where the inequality holds true, then write the solution as intervals reflecting these valid x-values.