Inequalities with multiplication

Learning Objectives Identify the inequality that represents the interior or exterior of a circle. Set up algebraic inequalities based on geometric conditions of circles, such as area constraints or point locations. Correctly apply the rule for multiplying or dividing an inequality by a negative number to find a solution. Solve for a variable that defines a circle's radius or other parameters within an inequality. Interpret the solution of an inequality in the geometric context of a circle's properties. Determine if a given point lies inside, on, or outside a region defined by a circle inequality. Ever wondered how a cell tower provides service within a certain 'zone' or how a sprinkler waters a circular lawn? 📡 These boundaries are defined by circles and...

TermDefinitionExample Equation of a CircleA formula that describes all points (x, y) on the edge of a circle. It is defined by its center (h, k) and its radius (r).The equation for a circle with its center at (2, -3) and a radius of 5 is `(x - 2)^2 + (y + 3)^2 = 25`. Interior of a CircleThe set of all points (x, y) located inside the boundary of a circle.For a circle defined by `x^2 + y^2 = 9`, the interior is the region where `x^2 + y^2 < 9`. Exterior of a CircleThe set of all points (x, y) located outside the boundary of a circle.For a circle defined by `x^2 + y^2 = 9`, the exterior is the region where `x^2 + y^2 > 9`. InequalityA mathematical statement that compares two values or expressions using symbols like < (less than), > (greater than), ≤ (less than or equal to), or ≥...

Circle Region Inequalities Interior: `(x - h)^2 + (y - k)^2 < r^2` Exterior: `(x - h)^2 + (y - k)^2 > r^2` Use these formulas to define the set of all points inside or outside a circle with center (h, k) and radius r. Use ≤ or ≥ to include the circle itself. The Negative Multiplication Rule If `a < b` and `c < 0`, then `ac > bc`. If `a > b` and `c < 0`, then `ac < bc`. This is the most critical rule for this topic. Whenever you multiply or divide both sides of an inequality by a negative number, you MUST flip the inequality symbol to maintain a true statement.