Problem 44 - Precalculus: Mathematics for Calculus

Question

$$\left\\{\begin{array}{r} x>0 \\ y>0 \\ x+y<10 \\ x^{2}+y^{2}>9 \end{array}\right.$$

Step-by-Step Solution

Answer

Points within the first quadrant, below \( x + y = 10 \), and outside the circle \( x^2 + y^2 = 9 \).

Step 1: Analyze the Inequalities

We have a system of four inequalities: 1. \( x > 0 \) which means \( x \) is a positive number. 2. \( y > 0 \) meaning \( y \) is also a positive number. 3. \( x + y < 10 \) which restricts the sum of \( x \) and \( y \) to be less than 10. 4. \( x^2 + y^2 > 9 \) indicating that the sum of their squares is more than 9.

Step 2: Understand the Region Defined by x + y < 10

The inequality \( x + y < 10 \) describes a region in the coordinate plane. This is a half-plane below the line \( x + y = 10 \), excluding the line itself. Since \( x > 0 \) and \( y > 0 \), we are interested in the first quadrant (where both \( x \) and \( y \) are positive).

Step 3: Visualize x^2 + y^2 > 9

The inequality \( x^2 + y^2 > 9 \) represents the region outside the circle centered at the origin with radius 3. Any point (\( x, y \)) satisfying this inequality lies strictly outside this circle.

Step 4: Find the Intersection of Regions

The solution to the system consists of points where all conditions are satisfied simultaneously: - Positive \( x \) and \( y \), - The line \( x + y = 10 \) creates a boundary, and - Points lie outside the circle with radius 3. By visualizing, notes that these conditions form a region that is in the first quadrant, below the 10 line, and outside the circle.

Key Concepts

Coordinate Plane

The coordinate plane is a central concept in algebra and geometry. It is a flat surface designed for plotting points, lines, and curves. You can think of it as a giant grid divided into four sections by two perpendicular lines called axes. These axes are labeled as the x-axis (horizontal) and the y-axis (vertical).

Here's how you use it:

• The intersection of the x-axis and y-axis is called the origin, often labeled as the point \( (0, 0) \).
• Coordinates are written as pairs \( (x, y) \) and indicate specific locations on the plane.
• The x-coordinate tells how far to move horizontally from the origin, while the y-coordinate tells how far to move vertically.

When working with inequalities, the coordinate plane helps visualize the solution sets. Different regions can be highlighted by shading, showing us where solutions to certain inequalities lie. For instance, the inequality \( x + y < 10 \) plots as a half-plane, with the line \( x + y = 10 \) forming a boundary, excluding the values on it.

Quadrant

The coordinate plane is divided into four parts, known as quadrants, each with its unique properties. Quadrants help categorize points based on their x and y coordinates:

• First Quadrant: Both x and y are positive, \( (x > 0, y > 0) \).
• Second Quadrant: x is negative while y is positive, \( (x < 0, y > 0) \).
• Third Quadrant: Both x and y are negative, \( (x < 0, y < 0) \).
• Fourth Quadrant: x is positive and y is negative, \( (x > 0, y < 0) \).

For problems involving inequalities, locating the relevant quadrant is key to visualizing solutions. In our exercise, since we work with \( x > 0 \) and \( y > 0 \), only the first quadrant is considered. Here, every point meets the criteria for both x and y being positive, guiding us in determining the area where our conditions all hold true.

Algebraic Inequalities

Algebraic inequalities are expressions involving variables, constants, and inequality signs like \( >, <, \leq, \geq \). They define ranges of values rather than exact numbers, offering flexibility in solutions.

Each inequality in a system contributes to the shape and extent of the solution space. Here's how they work in practice:

• Inequality \( x > 0 \): This simply indicates that x must be positive. Graphically, this covers the right side of the y-axis.
• Inequality \( y > 0 \): Similarly, y must be positive, covering the area above the x-axis.
• Inequality \( x + y < 10 \): This forms a boundary with points below the line \( x + y = 10 \).
• Inequality \( x^2 + y^2 > 9 \): This defines an area outside a circle of radius 3 centered at the origin, keeping solutions away from the origin.

By graphing these inequalities, you can identify regions where all conditions overlap, creating a visual representation of possible solutions. This is especially helpful for understanding complex systems where multiple inequalities interact.