Graph the solution set of each system of inequalities. x + 2y ≤ 4 y ≥ x² - 1

Verified step by step guidance

1. Step 1: Identify each inequality in the system. The first inequality is \(x - 4y \geq 2\) and the second inequality is \(y \geq x^{2} - 3\).
2. Step 2: For the first inequality \(x - 4y \geq 2\), rewrite it in terms of \(y\) to make graphing easier. Subtract \(x\) from both sides: \(-4y \geq 2 - x\). Then divide both sides by \(-4\), remembering to reverse the inequality sign because you are dividing by a negative number: \(y \leq \frac{x - 2}{4}\).
3. Step 3: For the second inequality \(y \geq x^{2} - 3\), recognize that this represents the region above or on the parabola \(y = x^{2} - 3\).
4. Step 4: Graph the boundary lines: the line \(y = \frac{x - 2}{4}\) (solid line because the inequality is inclusive) and the parabola \(y = x^{2} - 3\) (also solid line).
5. Step 5: Determine the solution set by shading the region below or on the line \(y = \frac{x - 2}{4}\) and above or on the parabola \(y = x^{2} - 3\). The solution set is where these shaded regions overlap.

Key Concepts

Graphing Linear Inequalities

Graphing linear inequalities involves plotting the boundary line given by the corresponding linear equation and then shading the region that satisfies the inequality. The boundary line is solid if the inequality includes equality (≥ or ≤) and dashed if it does not (> or <). For example, the inequality x - 4y ≥ 2 can be rewritten and graphed as a line, with shading above or below depending on the inequality.

Graphing Quadratic Inequalities

Graphing quadratic inequalities requires plotting the parabola defined by the quadratic equation y = x² - 3 and shading the region where the inequality holds. For y ≥ x² - 3, the region above or on the parabola is shaded. Understanding the shape and vertex of the parabola helps in accurately representing the solution set.

Solution Set of a System of Inequalities

The solution set of a system of inequalities is the intersection of the regions that satisfy each inequality individually. Graphically, this means finding the overlapping shaded area from each inequality's graph. This common region represents all points that satisfy all inequalities simultaneously.