In Exercises 27–62, graph the solution set of each system of inequalities or indicate that the system has no solution. y≥x^2−1, x−y≥−1

Key Concepts

Inequalities

Inequalities are mathematical expressions that show the relationship between two values when they are not equal. In this context, they indicate regions on a graph where certain conditions hold true, such as 'y is greater than or equal to a function of x'. Understanding how to interpret and graph inequalities is crucial for visualizing solution sets.

Graphing Quadratic Functions

Quadratic functions are polynomial functions of degree two, typically represented in the form y = ax² + bx + c. The graph of a quadratic function is a parabola, which can open upwards or downwards depending on the coefficient of x². In this problem, the inequality y ≥ (1/2)x² + 1 describes the region above the parabola, which is essential for determining the solution set.

Systems of Inequalities

A system of inequalities consists of two or more inequalities that are considered simultaneously. The solution set is the region where the graphs of the inequalities overlap. To solve such systems, one must graph each inequality and identify the common area that satisfies all conditions, which is critical for answering the given question.