In Exercises 27–62, graph the solution set of each system of inequalities or indicate that the system has no solution. y≥x²−1, x−y≥−1

Verified step by step guidance

1. Step 1: Understand the system of inequalities. The system is given as: $y\ge \frac{x^2}{2}+1$ and $4x\; -\; 2y\; \ge \; -2$.
2. Step 2: Graph the first inequality $y\; \ge \frac{x^2}{2}+\; 1$. This is a parabola opening upwards with vertex at (0,1). Since the inequality is ≥, shade the region above or on the parabola.
3. Step 3: Rewrite the second inequality $4x\; -\; 2y\; \ge \; -2$ in slope-intercept form to graph it easily. Start by isolating y: $-2y\; \ge \; -4x\; -\; 2$, then divide both sides by -2 (remember to reverse the inequality sign): $y\; \le \; 2x\; +\; 1$. This is a line with slope 2 and y-intercept 1. Shade the region below or on this line.
4. Step 4: Identify the solution set as the intersection of the two shaded regions: the area above the parabola and the area below the line.
5. Step 5: To better understand the solution set, find the points of intersection between the parabola and the line by solving the system: $y\; =\frac{x^2}{2}+\; 1$ and $y\; =\; 2x\; +\; 1$. Set them equal and solve for x: $\frac{x^2}{2}+\; 1\; =\; 2x\; +\; 1$. This will help determine the exact points where the shading regions meet.

Key Concepts

Graphing Quadratic Inequalities

Graphing quadratic inequalities involves plotting the parabola defined by the quadratic equation and shading the region that satisfies the inequality. For example, y ≥ (x²/2) + 1 means shading the area above or on the parabola y = (x²/2) + 1. Understanding the shape and direction of the parabola is essential.

Graphing Linear Inequalities

Linear inequalities like 4x - 2y ≥ -2 represent half-planes divided by the boundary line 4x - 2y = -2. To graph, first plot the boundary line, then determine which side satisfies the inequality by testing a point. The solution set includes all points on the boundary line and the shaded half-plane.

Solution Set of a System of Inequalities

The solution set of a system of inequalities is the intersection of the individual solution regions. It includes all points that satisfy every inequality simultaneously. Graphically, this is the overlapping shaded region from each inequality's graph, or no solution if the regions do not overlap.