Problem 26 - Precalculus

Question

Sketch the solution to each system of nonlinear inequalities in the plane. $$ \left\\{\begin{aligned} x^{2}+y^{2} & \geq 25 \\ y-x & \leq 1 \end{aligned}\right. $$

Step-by-Step Solution

Answer

The solution is the region outside the circle with radius 5 and below the line \(y = x + 1\).

Step 1: Understand the Inequalities

We have two inequalities to interpret. The first is \(x^2 + y^2 \geq 25\), which describes the region outside or on the boundary of a circle centered at the origin with radius 5. The second is \(y - x \leq 1\), which is the region below or on the line \(y = x + 1\).

Step 2: Graph the Circle

The inequality \(x^2 + y^2 \geq 25\) represents all points outside or on the perimeter of a circle centered at the origin \((0,0)\) with a radius of 5. Draw this circle on a coordinate plane and shade the area outside the circle.

Step 3: Graph the Line

The inequality \(y - x \leq 1\) can be rewritten as \(y \leq x + 1\). This is the region below or on the line that passes through the y-axis at (0,1) and has a slope of 1. Draw this line on the same coordinate plane and shade the region below it.

Step 4: Identify the Solution Region

The solution to the system of inequalities is the region where the shaded areas from both inequalities overlap. This region is outside the circle and below the line \(y = x + 1\).

Key Concepts

Coordinate Plane

Visualizing inequalities and their solutions requires the use of the coordinate plane. A coordinate plane is a two-dimensional surface where each point is defined by a pair of numerical coordinates. These coordinates, \(x, y\), are usually written in an ordered pair format.
The horizontal axis is known as the x-axis, while the vertical one is the y-axis. These axes intersect at a point called the origin, denoted as \(0,0\). Every point on the plane has a specific location based on these coordinates.

• For example, the point \(3,4\) is located 3 units to the right of the origin along the x-axis and 4 units up along the y-axis.
• Negative values locate points to the left (for x) and down (for y) from the origin.

It is crucial to have a strong understanding of the coordinate plane when working with equations and inequalities because it provides a visual representation of solutions.

System of Inequalities

A system of inequalities consists of multiple inequalities that need to be satisfied simultaneously. When dealing with inequalities, rather than finding a single point of intersection, we look for regions in the coordinate plane where all inequalities hold true.

• In this problem, the system includes two inequalities: \(x^2 + y^2 \geq 25\) and \(y - x \leq 1\).
• Each inequality represents a separate region on the coordinate plane.

Together, these inequalities describe a region that satisfies both conditions at once. This combined solution region is key to solving systems of inequalities.
It's like a Venn diagram overlapping where the shaded intersections show which solutions satisfy all the inequalities simultaneously.

Graphing Inequalities

Graphing inequalities involves several steps that transform algebraic conditions into visual areas on the coordinate plane. Let's highlight the two main inequalities from our exercise: the nonlinear inequality forming a circle and the linear inequality forming a line.\(\textbf{1. Graphing the Circle Inequality:}\)- The inequality \(x^2 + y^2 \geq 25\) corresponds to all points outside or on the boundary of a circle centered at the origin, with a radius of 5.

• Draw a circle with radius 5 around the origin.
• Shade the region outside the circle, indicating all potential solutions to this inequality.

\(\textbf{2. Graphing the Line Inequality:}\)- The inequality \(y - x \leq 1\) translates to \(y \leq x + 1\), a line with a slope of 1 passing through \((0,1)\).

• Draw this line on the plane.
• Shade the area below the line, showing all points that satisfy this condition.

Finally, the overlap of the shaded regions from both inequalities represents the solution to the system. This graphical representation is crucial to understanding systems of inequalities as it visually delineates the feasible region.