Solve each quadratic inequality. Give the solution set in interval notation. See Exam-ples 5 and 6. 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 \leq 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$.
3. Identify the roots from the previous step. These roots divide the number line into intervals. The roots are the critical points where the expression changes sign.
4. Test a value from each interval in the inequality \$3x^{2} + x - 4 \leq 0$ to determine whether the expression is less than or equal to zero in that interval.
5. Combine the intervals where the inequality holds true and express the solution set in interval notation, including endpoints if the inequality is 'less than or equal to' (\leq).

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 sign of the quadratic expression over different intervals.

Solving Quadratic Equations

To solve a quadratic inequality, first solve the related quadratic equation by setting the expression equal to zero. This finds critical points (roots) that divide the number line into intervals to test for the inequality's truth.

Interval Notation and Sign Analysis

After finding roots, use interval notation to express solution sets clearly. Test values from each interval to determine where the inequality holds true, then write the solution as a union of intervals where the quadratic expression meets the inequality condition.