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3.10 – Nonlinear Systems of Inequalities
Objectives
 Define nonlinear system of inequalities.
 Identify points that satisfy a nonlinear inequality.
 Match a nonlinear inequality with the graph that represents it.
 List the steps in the process of graphing a nonlinear inequality.
 Solve a nonlinear system of inequalities.
Key Terms
 Boundary Line – A solid or dashed line on a grpah that provides a border for the solution set.
 Solid: equal to (included in the solution)
 Dashed: not equal to (electric fence, do not cross, not included in the solution)
 Nonlinear Inequalities – Inequalities that are not straight (linear).
 Ex. Quadratic equations are nonlinear (they are parabolas)
Notes
Graphing Inequalities 
 The graph of this inequality is divided into two regions:
 Shaded: Shows all the points that are solutions
 Unshaded: Shows all the points that are not solutions
 How to Graph a Nonlinear Inequality
 Step 1: Change the inequality sign to an equal sign.
 Step 2: Graph the equation. Use dashed for < or >. Use solid for ≤ or ≥.
 Step 3: Choose a point (x,y) on either the inside or the outside the graph.
 Step 4: Substitute the point into the inequality and determine if that point makes the inequality true or false.
 Step 5: Shade
 If true, shade the region that contains that point.
 If false, shade the region that does not contain the point.


Examples 
 Example 1: Nonlinear Inequality System
 The region containing all the solutions for the circle inequality is blue.
 The regions containing all the solutions for the hyperbola inequality are pink.
 The regions containing all the solutions for the system of inequalities are purple.
 Every point inside the two purple regions makes both inequalities true.

Important!
Practice (Apex Study 3.10)
 Examples: Pg 6
 Practice: Pg 14, 15
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