Absolute Value Inequalities

Definition

Absolute value inequalities are mathematical expressions that involve the absolute value of a variable or expression being compared to a constant or another expression. These inequalities are used to find the range of values for the variable that satisfy the given inequality.

5 Must Know Facts For Your Next Test

1. Absolute value inequalities can be solved using the same basic steps as solving linear inequalities, but with the added step of considering the absolute value.
2. The solution set for an absolute value inequality can be expressed as the union of two intervals, one for when the expression inside the absolute value is positive and one for when it is negative.
3. The sign of the inequality may need to be reversed when the expression inside the absolute value is negative, depending on the type of inequality.
4. Graphing absolute value inequalities on a number line can help visualize the solution set and understand the relationship between the variable and the constant.
5. Absolute value inequalities can be used to model real-world situations, such as finding the range of acceptable values for a measurement or the distance between two points.

Review Questions

• Explain the process of solving an absolute value inequality, including the steps involved and the consideration of the sign of the inequality.
  • To solve an absolute value inequality, you first isolate the absolute value expression on one side of the inequality. Then, you consider the two possible cases: when the expression inside the absolute value is positive and when it is negative. For each case, you solve the resulting linear inequality, being careful to reverse the sign of the inequality if the expression inside the absolute value is negative. The final solution is the union of the two intervals obtained from the two cases.
• Describe how the graphing of an absolute value inequality on a number line can help in understanding the solution set.
  • Graphing an absolute value inequality on a number line can provide a visual representation of the solution set. The graph will consist of two intervals, one where the expression inside the absolute value is positive, and one where it is negative. The endpoints of these intervals correspond to the solutions to the inequality, and the graph can help you visualize the range of values that satisfy the inequality.
• Analyze how absolute value inequalities can be used to model real-world situations and explain the significance of the solution set in these applications.
  • Absolute value inequalities can be used to model a variety of real-world situations, such as finding the acceptable range of values for a measurement, the distance between two points, or the acceptable levels of a particular quantity. The solution set to the absolute value inequality represents the set of values that satisfy the given constraints, which is crucial for making informed decisions and understanding the limitations or requirements of the problem. The ability to interpret the solution set in the context of the real-world problem is an essential skill for applying absolute value inequalities effectively.

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. It is denoted by the symbol |x|.

Inequality:

An inequality is a mathematical statement that compares two expressions using symbols such as greater than (>), less than (<), greater than or equal to (≥), or less than or equal to (≤).

Number Line:

The number line is a visual representation of the set of real numbers, where each point on the line corresponds to a unique real number.