Graph the solution set of each system of inequalities.
$y\le x^3-x$
$y>-3$

Verified step by step guidance

1. Identify the two inequalities in the system: \(y \leq x^3 - x\) and \(y > -3\).
2. Graph the boundary curve for the first inequality, which is the cubic function \(y = x^3 - x\). Since the inequality is \(\leq\), include the curve itself as a solid line.
3. Shade the region below or on the curve \(y = x^3 - x\) because the inequality is \(y \leq x^3 - x\).
4. Graph the horizontal boundary line \(y = -3\). Since the inequality is strict (\(y > -3\)), draw this line as a dashed line to indicate points on the line are not included.
5. Shade the region above the line \(y = -3\) because the inequality is \(y > -3\). The solution set is the overlap of the shaded regions from both inequalities.

Key Concepts

Graphing Inequalities

Graphing inequalities involves shading the region of the coordinate plane that satisfies the inequality. For example, y ≤ x³ - x means shading all points on or below the curve y = x³ - x. The boundary curve is included if the inequality is ≤ or ≥, and excluded if it is < or >.

Cubic Functions and Their Graphs

A cubic function like y = x³ - x has an S-shaped curve with turning points. Understanding its shape helps in accurately sketching the boundary for the inequality. Key features include intercepts, local maxima and minima, and end behavior as x approaches ±∞.

Systems of Inequalities

A system of inequalities requires finding the region where all inequalities overlap. For y ≤ x³ - x and y > -3, the solution set is the area below the cubic curve but above the line y = -3. The final graph shows the intersection of these shaded regions.