Test Point Method

Definition

The test point method is a graphical technique used to determine the solution set of a linear inequality in two variables. It involves selecting a test point, evaluating the inequality at that point, and using the resulting sign to determine the region that satisfies the inequality.

5 Must Know Facts For Your Next Test

1. The test point method involves selecting a point in the coordinate plane and evaluating the inequality at that point to determine the region that satisfies the inequality.
2. The sign of the inequality at the test point (positive or negative) determines which half-plane contains the solution set.
3. If the inequality is $ax + by \geq c$, the solution set is the half-plane that contains the point where the inequality is true.
4. If the inequality is $ax + by > c$, the solution set is the half-plane that contains the point where the inequality is true, excluding the boundary line.
5. The test point method is a useful technique for graphing linear inequalities in two variables, as it allows you to visualize the solution set without having to solve the inequality algebraically.

Review Questions

• Explain the steps involved in using the test point method to graph a linear inequality in two variables.
  • To use the test point method to graph a linear inequality in two variables, follow these steps: 1) Identify the inequality, such as $2x + 3y \geq 6$. 2) Select a test point, such as (0, 0), and evaluate the inequality at that point. If the inequality is true at the test point, the solution set is the half-plane that contains the test point. If the inequality is false at the test point, the solution set is the other half-plane. 3) Draw the boundary line represented by the equation $2x + 3y = 6$. 4) Shade the appropriate half-plane to represent the solution set.
• Describe how the test point method can be used to determine the solution set of a strict linear inequality, such as $2x + 3y > 6$.
  • For a strict linear inequality, such as $2x + 3y > 6$, the test point method is used in a similar way. However, the key difference is that the solution set is the half-plane that contains the point where the inequality is true, excluding the boundary line. To use the test point method, you would 1) Identify the inequality, 2) Select a test point and evaluate the inequality, 3) Draw the boundary line represented by the equation $2x + 3y = 6$, and 4) Shade the appropriate half-plane, excluding the boundary line, to represent the solution set.
• Analyze how the test point method can be used to determine the intersection of the solution sets of multiple linear inequalities in two variables.
  • The test point method can be used to determine the intersection of the solution sets of multiple linear inequalities in two variables. To do this, you would 1) Graph each inequality using the test point method, 2) Identify the region where all the inequalities are satisfied, which is the intersection of the solution sets, and 3) Shade this common region to represent the final solution set. By using the test point method to graph each inequality individually, you can visualize the overall solution set that satisfies all the given linear inequalities.

Related terms

Linear Inequality: An inequality that can be written in the form $ax + by \geq c$ or $ax + by > c$, where $a$, $b$, and $c$ are real numbers.

Solution Set: The set of all points $(x, y)$ that satisfy a given linear inequality.

Boundary Line: The line represented by the equation $ax + by = c$ that separates the plane into two half-planes, one of which satisfies the inequality.