Multi-step inequalities

Learning Objectives Set up a multi-step inequality to determine if a point lies inside, on, or outside an ellipse. Solve quadratic inequalities to find the range of possible coordinates for a point on or within an elliptical region. Formulate and solve compound rational inequalities based on the eccentricity of an ellipse. Interpret the algebraic solution of an inequality in the geometric context of an ellipse. Determine the valid range for a variable in the definition of an ellipse's parameters (e.g., semi-major axis, focal distance). Distinguish between strict (<, >) and non-strict (≤, ≥) inequalities when describing regions relative to an ellipse. How can engineers determine the 'safe zone' for a satellite in an elliptical orbit to avoid space debris?...

TermDefinitionExample Elliptical BoundaryThe set of all points (x, y) that satisfy the standard equation of an ellipse. This is the curve of the ellipse itself.For the ellipse given by (x^2/25) + (y^2/9) = 1, the point (5, 0) is on the boundary because (5^2/25) + (0^2/9) = 1 + 0 = 1. Interior of an EllipseThe set of all points (x, y) located inside the elliptical boundary. These points satisfy the ellipse equation with the '=' replaced by a '<' sign.The point (1, 1) is in the interior of (x^2/25) + (y^2/9) = 1 because (1^2/25) + (1^2/9) = 1/25 + 1/9 ≈ 0.15, which is less than 1. Exterior of an EllipseThe set of all points (x, y) located outside the elliptical boundary. These points satisfy the ellipse equation with the '=' replaced by a '>' si...

Elliptical Region Formulas (Horizontal Major Axis) Given the ellipse \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1: Interior: \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} < 1 Exterior: \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} > 1 Use these inequalities to test if a point is inside or outside an ellipse. Substitute the point's coordinates (x, y) into the expression and compare the result to 1. Eccentricity Inequality For any ellipse, 0 < e < 1, where e = \frac{c}{a} = \frac{\sqrt{a^2 - b^2}}{a}. This compound inequality must always be true for a shape to be an ellipse. Use it to find the valid range of values for a variable that defines the ellipse's parameters a, b, or c.