Solve each quadratic inequality. Give the solution set in interval notation. See Exam-ples 5 and 6. 3x^2+x≤4

Quadratic inequalities are expressions that involve a quadratic polynomial set in relation to a value, typically using symbols like ≤, ≥, <, or >. To solve these inequalities, one must first rearrange the equation to set it to zero, then find the roots of the corresponding quadratic equation. The solution involves determining the intervals where the quadratic expression is either less than or greater than zero.

Interval notation is a mathematical notation used to represent a range of values. It uses parentheses and brackets to indicate whether endpoints are included (closed intervals) or excluded (open intervals). For example, the interval (2, 5] includes all numbers greater than 2 and up to 5, including 5 but not 2. This notation is essential for expressing the solution set of inequalities clearly.

Graphing quadratics involves plotting the quadratic function on a coordinate plane to visualize its shape, which is a parabola. The vertex, axis of symmetry, and intercepts are key features that help in understanding the behavior of the quadratic. By analyzing the graph, one can determine the intervals where the quadratic is above or below the x-axis, which directly relates to solving quadratic inequalities.