### Shortcut Methods

**Shortcut Methods and Tricks to Solve Linear Inequality Numericals**

**1.** To graph a linear inequality, first rewrite the inequality in slope-intercept form (y = mx + b), then graph the line represented by the equation. The region that satisfies the inequality will be on one side of the line, and the other side of the line will be the region that does not satisfy the inequality.

**2.** To find the vertices of the feasible region for a system of linear inequalities, first graph each inequality individually. The feasible region is the region that is common to all of the graphs. The vertices of the feasible region are the points where two or more of the graphs intersect.

**3.** To find the maximum and minimum values of a function in a feasible region, first find the critical points of the function. The critical points are the points where the first derivatives of the function are zero or undefined. Then, evaluate the function at the critical points and at the vertices of the feasible region. The maximum value of the function is the largest of these values, and the minimum value of the function is the smallest of these values.

**4.** To prove that the set of points satisfying the linear inequality ax + by ≤ c is a closed half-plane, you can use the following steps:

- First, show that the set of points satisfying the inequality is a half-plane. This can be done by showing that the inequality defines a line and that the region on one side of the line satisfies the inequality.
- Next, show that the set of points satisfying the inequality is closed. This can be done by showing that every limit point of the set is in the set.

**5.** To find the area of the region enclosed by a system of linear inequalities, you can use the following steps:

- First, graph the system of inequalities.
- Then, divide the region into smaller shapes, such as triangles and rectangles.
- Finally, find the area of each shape and add them together to find the total area of the region.

**6.** To find the minimum value of a function in a feasible region, you can use the following steps:

- First, find the critical points of the function.
- Then, evaluate the function at the critical points and at the vertices of the feasible region.
- The minimum value of the function is the smallest of these values.

**7.** To graph a linear inequality, first rewrite the inequality in slope-intercept form (y = mx + b), then graph the line represented by the equation. The region that satisfies the inequality will be on one side of the line, and the other side of the line will be the region that does not satisfy the inequality.

**8.** To find the vertices of the feasible region for a system of linear inequalities, first graph each inequality individually. The feasible region is the region that is common to all of the graphs. The vertices of the feasible region are the points where two or more of the graphs intersect.

**9.** To find the maximum and minimum values of a function in a feasible region, first find the critical points of the function. The critical points are the points where the first derivatives of the function are zero or undefined. Then, evaluate the function at the critical points and at the vertices of the feasible region. The maximum value of the function is the largest of these values, and the minimum value of the function is the smallest of these values.

**10.** To solve a linear programming problem, you can use the following steps:

- First, graph the feasible region.
- Then, find the corner points of the feasible region.
- Finally, evaluate the objective function at each corner point. The corner point that gives the largest value of the objective function is the optimal solution to the problem.