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. Rewrite the inequality in standard form by moving all terms to one side: \(3x^{2} + x - 4 \geq 0\).
2. Factor the quadratic expression \$3x^{2} + x - 4\( if possible, or use the quadratic formula to find its roots. The quadratic formula is given by \(x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\), where \)a=3\(, \)b=1\(, and \)c=-4$.
3. Identify 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 determine where the inequality holds true.
5. Write the solution set in interval notation based on the intervals where the inequality is satisfied, including endpoints if the inequality is 'greater than or equal to'.

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, setting the quadratic expression to zero, and determining where the parabola lies above or below the x-axis.

Factoring and Finding Roots

Factoring the quadratic expression or using the quadratic formula helps find the roots (zeros) of the quadratic equation. These roots divide the number line into intervals, which are tested to determine where the inequality holds true.

Interval Notation

Interval notation is a concise way to express the solution set of inequalities. It uses parentheses and brackets to indicate open or closed intervals, representing all values that satisfy the inequality on the number line.