2.3 Linear Inequalities - Notes PDF

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2.3 LINEAR INEQUALITIES A linear inequality is a linear equation where the equals sign is replaced by an inequality symbol.

• Standard form of a linear inequality in two variables (a, b, and c are constants)

• Standard form of a linear inequality in n variables (a1 , a2 , ..., an , c are constants)

A solution to a linear inequality is a point that satisfies the inequality.

• Solution to a linear inequality in two variables x and y

• Solution to a linear inequality in n variables x1 , x2 , ..., xn

For example, (1, 0) is a solution to the inequality 4x + y > 1 since:

The set of all solutions to a linear inequality can be represented graphically, as shown below:

1

Example. Sketch the regions determined by each of the following inequalities. 1 1. x − y ≤ 4 3

2. 4y ≥ 10

3. x + 2y ≤ 1

4. x ≥ 0

A system of linear inequalities is a set of two or more linear inequalities to be satisfied simultaneously. Example of a system of linear inequalities in two variables:

A solution to a system of linear inequalities is a point that is a solution to each individual linear inequality. For example, (4, 1) is a solution to the example system of linear inequalities since:

The set of all solutions to a system of inequalities is called the feasible region. Graphically, the feasible region is the shared ”white area” leftover after shading each individual inequality. A feasible region is either bounded or unbounded. • bounded

• unbounded

The intersections of the boundaries of the feasible region are called corner points. Example. Sketch the feasible region that corresponds to the given set of inequalities. Decide whether the feasible region is bounded or unbounded. Find the coordinates of all corner points (if any). 3x + 2y ≥ 6 3x − 2y ≤ 6 x≥0

Example. Sketch the feasible region that corresponds to the given set of inequalities. Decide whether the feasible region is bounded or unbounded. Find the coordinates of all corner points (if any). x + 3y ≤ 9 2x − y ≥ −1 x ≥ 0, y ≥ 0

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