Solve each quadratic inequality. Give 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 \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$.
3. Determine the roots from the quadratic formula or factoring. These roots divide the number line into intervals.
4. Test a value from each interval in the inequality \$3x^{2} + x - 4 \leq 0$ to see if the inequality holds true in that interval.
5. Based on the test results, write the solution set in interval notation, including the roots if the inequality is less than or equal to zero (since equality is allowed).

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 its graph.

Solving Quadratic Equations

To solve a quadratic inequality, first solve the corresponding quadratic equation by setting it equal to zero. This helps find critical points (roots) that divide the number line into intervals to test for the inequality.

Interval Notation and Testing Intervals

After finding the roots, the number line is split into intervals. Each interval is tested in the inequality to determine if it satisfies the condition. The solution set is then expressed using interval notation, which concisely represents all valid x-values.