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3.10 – Nonlinear Systems of Inequalities
- 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.
- 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)
- 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.
- 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.
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