Problem 47 - Precalculus: Functions and Graphs

Question

Graph the system of inequalities. $$\left\\{\begin{array}{c}x^{4}-2 x<3 y \\\x+2 y

Step-by-Step Solution

Answer

Graph each inequality as a boundary curve and shade the overlapping region.

Step 1: Rewrite the Inequalities

First, we need to rewrite each inequality to make them easier to graph. Start with the first inequality:\[ x^4 - 2x < 3y \]Divide both sides by 3 to solve for \(y\):\[ y > \frac{x^4 - 2x}{3} \]Now, take the second inequality:\[ x + 2y < x^3 - 5 \]Subtract \(x\) from both sides and solve for \(y\):\[ 2y < x^3 - x - 5 \]Divide by 2:\[ y < \frac{x^3 - x - 5}{2} \]

Step 2: Graph the Boundary Lines

Let's graph the boundary lines for each inequality. Consider the equations as equalities first:1. \( y = \frac{x^4 - 2x}{3} \) - This is a polynomial curve.2. \( y = \frac{x^3 - x - 5}{2} \) - This is another polynomial curve.Graph these equations on a coordinate plane. Since these are nonlinear equations, their graphs will resemble curves rather than straight lines.

Step 3: Determine the Regions for the Inequalities

Since the inequalities are strict (\(<\) and \(>\)), we will use dashed lines for the boundaries of the regions:- For \(y > \frac{x^4 - 2x}{3}\), shade the region above the curve.- For \(y < \frac{x^3 - x - 5}{2}\), shade the region below the curve.The solution will be the overlapping shaded region for both inequalities.

Step 4: Identify the Solution Area

Examine where the shaded regions overlap. The area where the shading from both inequalities intersects represents the solution to the system of inequalities. This region satisfies both inequalities.

Key Concepts

Polynomial Inequalities

Polynomial inequalities involve expressions where the highest power of the variable is greater than one, often creating curves when graphed. When solving polynomial inequalities like \(x^4 - 2x < 3y\) and \(x + 2y < x^3 - 5\), it is crucial to understand how these inequalities behave.

• The expressions on each side of the inequality are typically polynomials, which means they include terms with variables raised to a power (e.g., \(x^4\), \(x^3\)).
• Unlike linear equations, polynomial equations can have multiple turning points, creating graphs that are not straight lines, but curves.
• For solving inequalities, it is often useful to rearrange them by isolating one variable, usually solving for \(y\) to make graphing easier.

This process involves dividing or otherwise rearranging terms to give a function \(y = f(x)\) which can be more easily plotted on a graph.

Coordinate Plane

Graphing on a coordinate plane is a visual way to represent algebraic equations and inequalities. The coordinate plane consists of two perpendicular lines, the x-axis, and the y-axis, which divide the plane into four quadrants.

• Each point can be represented by an ordered pair \((x, y)\), where \(x\) is the horizontal value and \(y\) is the vertical value.
• When graphing the boundary lines of an inequality, the equations are initially treated as if they were equalities (e.g., \( y = \frac{x^4 - 2x}{3} \)).
• The intersecting curves or lines will represent potential solutions to the inequalities before considering which regions meet the inequalities' criteria.

By utilizing the coordinate plane effectively, one can visually determine where solutions lie, making it a crucial tool for solving systems of inequalities.

Shaded Regions

Shaded regions on a graph represent the set of points that satisfy an inequality. Once the boundary lines are graphed, finding the correct shaded region involves understanding the inequality's orientation.

• For inequalities like \( y > \frac{x^4 - 2x}{3} \), the region above the curve is shaded because any point \((x, y)\) above will satisfy the inequality.
• Conversely, for \( y < \frac{x^3 - x - 5}{2} \), shading occurs below the curve.
• It's important to remember that dashed lines on the graph indicate that the boundary is not included in the solution set (as the inequality is strict in this case).

The area where the shading from both inequalities overlap is crucial. This common shaded region represents the solutions that satisfy both inequalities simultaneously. Identifying and shading these regions correctly helps in visualizing the solution set for the system of inequalities.