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

Verified step by step guidance

1. Graph the inequality $y \geq \frac{x^2}{2} + 1$. This is a parabola opening upwards with vertex at $(0, 1)$. Shade the region above the parabola.
2. Graph the inequality $4x - 2y \geq -2$. First, rewrite it in slope-intercept form: $y \leq 2x + 1$. This is a line with slope 2 and y-intercept 1. Shade the region below the line.
3. Identify the intersection of the shaded regions from both inequalities. This is the solution set of the system.
4. Check if the boundary lines are included in the solution set. Since both inequalities are 'greater than or equal to' or 'less than or equal to', the boundary lines are included.
5. Verify the solution by selecting a test point from the intersection region and checking if it satisfies both inequalities.

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.