91Ó°ÊÓ

Inequalities describe a range of possible values for a set of variables. In this exercise, we look at the inequality \( x^2 + y^2 \leq 1 \). This tells us that for any point \((x, y)\), the sum of the squares of x and y must be less than or equal to 1.
This inequality includes two types of points:

• Points on the boundary, where \( x^2 + y^2 = 1 \). This is where the 'equals' part of the inequality applies.
• Points inside the boundary, where \( x^2 + y^2 < 1 \). These points have a sum of squares that is strictly less than the boundary value.

When dealing with such inequalities involving two variables, it often forms shapes on a coordinate plane. Here, the inequality is specific to a circular region, which leads us to understand circle equations next.

Circle equations typically appear in the form \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the circle's center and \(r\) its radius.
In our case:

• The equation is \(x^2 + y^2 = 1\), indicating a circle.
• The center is at the origin \((0,0)\), because there are no \(h\) or \(k\) shifts.
• The radius is 1, taken from the square root of 1, which is \(r^2\) in the equation.

The inequality \(x^2 + y^2 \leq 1\) means that all points that satisfy the circle equation and those contained within the circle will satisfy this inequality. This understanding is crucial for graphing the circle on a coordinate plane.

Plotting the equation \(x^2 + y^2 = 1\) is an essential graphing technique that helps visualize inequalities involving circles. Begin with identifying intersection points on the axes:

• The circle intersects the x-axis at \(+1\) and \(-1\).
• It intersects the y-axis at \(+1\) and \(-1\).

After marking these points, connect them with a smooth curve to sketch the circle.
To represent the inequality \(x^2 + y^2 \leq 1\), shade the area within the circle. This shaded portion includes everything inside the circle, showing visually that the circle's interior and boundary satisfy the inequality.
Shading serves to reinforce what the inequality \( \leq 1 \) conveys, which includes all points either on or inside the circle, and this method makes differences between simple equations and inequalities clear in visual form.