Isolation Method

Definition

The isolation method is a technique used to solve absolute value inequalities by isolating the variable on one side of the inequality. This allows for the application of the properties of absolute values to find the solution set.

5 Must Know Facts For Your Next Test

1. The isolation method involves manipulating the absolute value inequality to isolate the variable on one side, often by using the properties of absolute values.
2. Once the variable is isolated, the inequality can be solved by considering the cases where the expression inside the absolute value is positive and where it is negative.
3. The solution set for an absolute value inequality can consist of one or two intervals, depending on the specific inequality and the values that satisfy it.
4. The isolation method is a crucial step in solving absolute value inequalities, as it allows for the application of the properties of absolute values to find the final solution set.
5. Mastering the isolation method is essential for successfully solving a wide range of absolute value inequalities that may appear on tests and in real-world applications.

Review Questions

• Explain the purpose of the isolation method in the context of solving absolute value inequalities.
  • The isolation method is used to solve absolute value inequalities by isolating the variable on one side of the inequality. This is a crucial step because it allows for the application of the properties of absolute values, which are necessary to determine the solution set. By isolating the variable, the inequality can be rewritten in a form that can be more easily analyzed and solved.
• Describe the process of using the isolation method to solve an absolute value inequality, such as $|x - 3| < 5$.
  • To solve the absolute value inequality $|x - 3| < 5$ using the isolation method, we would first isolate the variable $x$ by subtracting 3 from both sides of the inequality: $|x - 3| < 5$. This gives us $x - 3 < 5$ and $x - 3 > -5$. Next, we would consider the two cases: when $x - 3$ is positive and when $x - 3$ is negative. Solving these cases separately, we would find that the solution set is $-2 < x < 8$.
• Analyze the relationship between the isolation method and the properties of absolute values, and explain how this relationship is crucial for solving absolute value inequalities.
  • The isolation method and the properties of absolute values are intrinsically linked when solving absolute value inequalities. By isolating the variable on one side of the inequality, we can then apply the properties of absolute values, such as $|x| = |-x|$ and $|x| \geq 0$, to rewrite the inequality in a form that can be more easily solved. This relationship is crucial because the properties of absolute values are necessary to determine the solution set, and the isolation method provides the foundation for applying these properties effectively. The combination of isolating the variable and using the absolute value properties is the key to successfully solving a wide range of absolute value inequalities.

Related terms

Absolute Value Inequality: An inequality that involves the absolute value of a variable or expression, such as $|x - 3| < 5$.

Solution Set: The set of all values of the variable that satisfy the given inequality.

Absolute Value Properties: The mathematical properties that govern the behavior of absolute values, such as $|x| = |-x|$ and $|x| \geq 0$.