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A system of inequalities involves multiple inequality conditions that must be satisfied simultaneously. In our example, we look at a system involving two quadratic inequalities:

• \(y \leq -x^2 + 5\)
• \(y \leq x^2 - 3\)

To solve this, we graph each inequality and find the region where their solution sets overlap. When two inequalities are present, finding their solution requires analyzing where both conditions hold true in the same space.
The steps to approach such problems involve graphing their respective boundary lines, examining the graphs to determine the feasible regions, and then considering where these regions intersect. This comprehensive method ensures that the solution set considers all given inequalities simultaneously.

In our problem, the inequalities contain parabolas, which are curves described by quadratic equations. Here's how to handle them:.

• For the inequality \(y \leq -x^2 + 5\), we start by graphing the parabola \(y = -x^2 + 5\), a downward-opening parabola with a vertex at (0, 5).
• For the inequality \(y \leq x^2 - 3\), we graph the parabola \(y = x^2 - 3\), an upward-opening parabola with a vertex at (0, -3).

When graphing these parabolas, use a solid line to indicate that points on the line satisfy the inequality (this is because the inequality sign includes \(\leq\), meaning 'less than or equal to').The shape and direction of these parabolas come from the coefficients of the \(x^2\) term:.

• A positive coefficient means the parabola opens upwards.
• A negative coefficient means it opens downwards.

After graphing the parabolas, the next crucial step is to shade the appropriate regions that satisfy each inequality. Here's what to do:.

• For \(y \leq -x^2 + 5\), shade the region below or on the downward parabola \(y = -x^2 + 5\).
• For \(y \leq x^2 - 3\), shade the region below or on the upward parabola \(y = x^2 - 3\).

The solution to the system of inequalities is the overlapping area where the shaded regions from both inequalities intersect. This shared area represents all the points (x, y) that satisfy both inequality conditions simultaneously. To ensure clarity and correctness, carefully check the shaded regions to confirm they meet all criteria outlined by the inequalities.