Use the technique described in Exercises 87–90 to solve each inequality. Write the solution set in interval notation. 3x^2 + x ≥ 4

Verified step by step guidance

1. Rewrite the inequality in standard form: $3x^2 + x - 4 \geq 0$.
2. Factor the quadratic expression $3x^2 + x - 4$ if possible, or use the quadratic formula to find the roots.
3. Determine the critical points by setting the expression equal to zero: $3x^2 + x - 4 = 0$.
4. Use the critical points to divide the number line into intervals and test each interval to determine where the inequality holds true.
5. Write the solution set in interval notation, including endpoints where the inequality is satisfied.

Key Concepts

Inequalities

Inequalities are mathematical statements that express the relationship between two expressions that are not necessarily equal. They use symbols such as '≥' (greater than or equal to) and '≤' (less than or equal to) to indicate the direction of the relationship. Understanding how to manipulate and solve inequalities is crucial for finding solution sets.

Quadratic Functions

A quadratic function is a polynomial function of degree two, typically expressed in the form f(x) = ax^2 + bx + c. The graph of a quadratic function is a parabola, which can open upwards or downwards depending on the sign of the leading coefficient 'a'. Solving inequalities involving quadratics often requires finding the roots of the equation and analyzing the intervals defined by these roots.

Interval Notation

Interval notation is a mathematical notation used to represent a range of values on the number line. It uses parentheses and brackets to indicate whether endpoints are included (closed interval) or excluded (open interval). For example, the interval [a, b) includes 'a' but not 'b', which is essential for expressing solution sets of inequalities clearly and concisely.