Calculating the Area of the Region Bounded by Inequalities

The question asks us to find the area of a region defined by the inequalities and equations: |x| < 5, y = 0, and y = 8. To solve this, we first need to understand what these conditions represent graphically.

Understanding the Boundaries of the Region

• The inequality |x| < 5 means that the absolute value of x is less than 5. This translates to -5 < x < 5. Graphically, this represents the region between the vertical lines x = -5 and x = 5.
• The equation y = 0 represents the x-axis.
• The equation y = 8 represents a horizontal line parallel to the x-axis, located 8 units above it.

Combining these conditions, the region is bounded by the vertical lines x = -5 and x = 5, and the horizontal lines y = 0 and y = 8.

Identifying the Shape of the Region

A region bounded by two vertical lines (x = constant) and two horizontal lines (y = constant) forms a rectangle. In this case, the vertices of the rectangle are at the intersections of these boundary lines:

• (-5, 0)
• (5, 0)
• (5, 8)
• (-5, 8)

Calculating the Dimensions of the Rectangle

To find the area of the rectangle, we need its width and height.

• Width: The distance between the two vertical boundaries, x = -5 and x = 5. This distance is given by 5 - (-5) = 5 + 5 = 10 units.
• Height: The distance between the two horizontal boundaries, y = 0 and y = 8. This distance is given by 8 - 0 = 8 units.

Calculating the Area

The area of a rectangle is calculated by multiplying its width by its height.

Area = Width \(\times\) Height

Area = \(10 \times 8\)

Area = 80 square units.

Therefore, the area of the region bounded by |x| < 5, y = 0, and y = 8 is 80 square units.

Revision Table: Key Concepts for Area Calculation

Additional Information: Regions Bounded by Inequalities

Understanding how to interpret inequalities is crucial for finding the area of bounded regions. Inequalities like |x| < a, |x| > a, |y| < b, or |y| > b define bands or regions between or outside certain vertical or horizontal lines.

• |x| < a: Region between x = -a and x = a.
• |x| > a: Region to the left of x = -a and to the right of x = a.
• |y| < b: Region between y = -b and y = b.
• |y| > b: Region below y = -b and above y = b.

Combining these with linear equations (like y = mx + c or x = constant, y = constant) helps define specific geometric shapes (rectangles, triangles, trapezoids, etc.) or more complex regions. The area of such regions can often be found using basic geometric formulas or, for more complex cases, using integration in calculus.