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### Vocabulary

##### Complete the chart.

Word | Definition |

________________ | the part on the graph where the shaded areas of the inequalities overlap |

Feasible Region | _________________________________________________________ |

________________ | a restriction or condition presented in a real-world problem |

Linear Programming | _________________________________________________________ |

Vertex Theorem for Regions | _________________________________________________________ |

### Linear Inequalities

The first step of graphing linear inequalities is to write the inequality in slope-intercept form.

What is the form that linear inequalities are typically written in? ____________________

What is slope-intercept form? ____________________

*Remember:**When an inequality is divided or multiplied by a negative number, the sign is reversed.*

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Complete the following chart.

Inequality Symbol | In Words | Dashed or solid? |

less than | solid | |

___________________ | ___________________ | |

___________________ | ___________________ | |

___________________ | ___________________ |

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Once you graph the line, you must decide where to shade. Choose a point not on the line and plug it in. If it is a solution, you shade that area. If not, shade the other area.

*Hint:**Once graphed, find the y-intercept. If the equation is or then shade the region under the y-intercept, if or shade the region above the y-intercept.*

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Without graphing, determine if each point is in the shaded region for each inequality.

- (–1, 3) and
- (–5, –1) and
- (6, 2) and

Determine the inequality that is modeled by each of the following graphs.

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#### Graphs of Systems of Linear Inequalities

To graph systems of linear inequalities, graph each inequality separately and find where their shaded regions intersect. This region of intersection is called the ______________________.

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For each of the following systems of inequalties, a) solve by graphing, and b) determine three points that satisfy the system.

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#### Applications of Systems of Inequalities

A system of linear inequalities is often used to determine the maximum or minimum values of a situation with multiple constraints.

In order to solve this type of problem using linear inequalities, follow these steps:

- Make a table to organize the given information.
- List the constraints of the situation. Write an inequality for each constraint.
- Write an equation for the quantity you are trying to maximize (like profit) or minimize (like cost).
- Graph the constraints as a system of inequalities.
- Find the exact coordinates for each vertex from the graph or algebraically.
- Use the Vertex Theorem. Test all vertices of the feasible region in the equation and see which point is the maximum or minimum.

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For each graphed region and corresponding equation, find a point at which ‘ ’ has a maximum value and a point at which ‘ ’ has a minimum value.

Beth is knitting mittens and gloves. Each pair must be processed on three machines. Each pair of mittens requires 2 hours on Machine A, 2 hours on Machine B and 4 hours on Machine C. Each pair of gloves requires 4 hours on Machine A, 2 hours on Machine B and 1 hour on Machine C. Machine A, B, and C are available 32, 18 and 24 minutes each day respectively. The profit on a pair of mittens is $8.00 and on a pair of gloves is $10.00. How many pairs of each should be made each day to maximize the profit?

- List the constraints and state the profit equation.
- Create a graph and identify the feasible region.
- Determine what the Beth must do to maximize her profit.

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