Tolerance

Definition

Tolerance refers to the ability to withstand or endure a certain level of a substance, condition, or stimulus without adverse effects. In the context of solving absolute value inequalities, tolerance is a crucial concept that determines the range of values that satisfy the inequality.

5 Must Know Facts For Your Next Test

1. Tolerance in the context of absolute value inequalities determines the range of values that satisfy the inequality.
2. The tolerance value represents the maximum or minimum distance from the reference value that is considered acceptable.
3. Solving absolute value inequalities involves finding the solution set, which consists of all values that fall within the tolerance range.
4. The tolerance value is often represented by a variable, such as $k$, and the inequality is expressed in the form $|x - a| \leq k$ or $|x - a| \geq k$.
5. The solution set of an absolute value inequality can be determined by considering the cases where the expression inside the absolute value is positive or negative.

Review Questions

• Explain how the tolerance value affects the solution set of an absolute value inequality.
  • The tolerance value, represented by $k$, determines the range of values that satisfy the absolute value inequality. If the inequality is in the form $|x - a| \leq k$, the solution set consists of all values of $x$ that fall within the interval $[a - k, a + k]$. Conversely, if the inequality is in the form $|x - a| \geq k$, the solution set consists of all values of $x$ that fall outside the interval $[a - k, a + k]$. The tolerance value, therefore, plays a crucial role in defining the boundaries of the solution set for absolute value inequalities.
• Describe the process of solving an absolute value inequality with a given tolerance value.
  • To solve an absolute value inequality with a given tolerance value, $k$, the following steps are typically followed: 1. Rewrite the inequality in the standard form: $|x - a| \leq k$ or $|x - a| \geq k$, where $a$ is the reference value. 2. Consider the cases where the expression inside the absolute value is positive and negative. 3. For the case where the expression is positive, solve the inequality $x - a \leq k$ or $x - a \geq k$. 4. For the case where the expression is negative, solve the inequality $-(x - a) \leq k$ or $-(x - a) \geq k$. 5. Combine the solutions from both cases to obtain the final solution set.
• Analyze the relationship between the tolerance value and the size of the solution set for an absolute value inequality.
  • The tolerance value, $k$, has a direct impact on the size of the solution set for an absolute value inequality. When the inequality is in the form $|x - a| \leq k$, a larger tolerance value $k$ will result in a wider solution set, as the range of acceptable values increases. Conversely, a smaller tolerance value $k$ will lead to a narrower solution set. For the inequality $|x - a| \geq k$, a larger tolerance value $k$ will result in a smaller solution set, as the range of unacceptable values increases. Understanding the relationship between the tolerance value and the size of the solution set is crucial in solving absolute value inequalities and interpreting the results.

Related terms

Absolute Value: The absolute value of a number is the distance of that number from zero on the number line, regardless of its sign.

Inequality: An inequality is a mathematical statement that compares two expressions and indicates that one is greater than, less than, or not equal to the other.

Solution Set: The solution set of an inequality is the set of all values that make the inequality true.