Use the technique described in Exercises 87–90 to solve each inequality. Write the solution set in interval notation. 3x² + x ≥ 4

Verified step by step guidance

1. First, rewrite the inequality so that one side is zero: \$3x^{2} + x - 4 \geq 0$.
2. Next, factor the quadratic expression \$3x^{2} + x - 4\( if possible, or use the quadratic formula to find its roots. The quadratic formula is \)x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\(, where \)a=3\(, \)b=1\(, and \)c=-4$.
3. Determine the critical points (roots) from the previous step. These points divide the number line into intervals.
4. Test a value from each interval in the inequality \$3x^{2} + x - 4 \geq 0$ to see if the inequality holds true in that interval.
5. Based on the test results, write the solution set in interval notation, including endpoints where the inequality is equal (since it is \(\geq\) and not just >).

Key Concepts

Solving Quadratic Inequalities

Solving quadratic inequalities involves finding the values of the variable that make the inequality true. This typically requires rewriting the inequality in standard form, factoring or using the quadratic formula to find critical points, and then testing intervals to determine where the inequality holds.

Interval Notation

Interval notation is a way to represent sets of numbers on the number line. It uses parentheses () for values not included and brackets [] for values included, describing continuous ranges such as (a, b), [a, b), or (-∞, c]. This notation concisely expresses solution sets of inequalities.

Sign Analysis of Quadratic Expressions

Sign analysis involves determining where a quadratic expression is positive, negative, or zero by examining the intervals defined by its roots. After finding the roots, test points in each interval to see if the expression satisfies the inequality, helping to identify the solution set.