Solving an inequality with an absolute value requires a different approach than a standard inequality. Because absolute value represents distance from zero, these problems often result in a compound solution—either a bounded range of numbers or two separate, unbounded ranges. Our Absolute Value Inequalities Calculator helps students navigate this tricky algebraic concept. It solves for the range of values that satisfy the inequality and presents the answer in both inequality and interval notation, making it an excellent tool for learning and checking homework.
How to Use the Absolute Value Inequalities Calculator
Finding the solution set for your inequality is a simple process:
- Enter the Coefficients: Input the numerical values for 'a', 'b', and 'c' from your inequality.
- Select the Operator: Choose the correct inequality symbol from the dropdown menu (<, ≤, >, or ≥).
- Solve the Inequality: Click the "Solve" button.
- View the Solution: The calculator will display the solution set as a compound inequality and in standard interval notation.
Understanding the Two Types of Absolute Value Inequalities
The key to solving absolute value inequalities is to recognize which of the two fundamental cases you are dealing with. The inequality symbol is the deciding factor.
Case 1: Less Than (< or ≤) - The "And" Statement
When you have an inequality like |x| < 5, it's asking, "What numbers have a distance from zero that is *less than* 5?" The numbers that satisfy this are all the numbers between -5 and 5. This translates into a compound "and" statement:
-5 < x < 5
This is a bounded interval. When you have a more complex expression, like |ax + b| ≤ c, you break it into a similar compound inequality and solve for x:
-c ≤ ax + b ≤ c
Case 2: Greater Than (> or ≥) - The "Or" Statement
When you have an inequality like |x| > 5, it's asking, "What numbers have a distance from zero that is *greater than* 5?" The numbers that satisfy this fall into two separate groups: all the numbers greater than 5, OR all the numbers less than -5. This translates into a compound "or" statement:
x > 5 or x < -5
This represents two unbounded intervals. For a complex expression like |ax + b| ≥ c, you must set up and solve two separate inequalities:
ax + b ≥ c OR ax + b ≤ -c
Interval Notation: A Quick Guide
Interval notation is a standardized way to represent a set of numbers. It uses parentheses and brackets to indicate whether the endpoints are included.
- Parentheses ( ): Used when an endpoint is *not* included in the solution set. This corresponds to the < and > symbols.
- Brackets [ ]: Used when an endpoint *is* included in the solution set. This corresponds to the ≤ and ≥ symbols.
- Infinity (∞): Since you can never actually reach infinity, it is always used with a parenthesis.
- Union Symbol (U): This symbol is used to connect two separate, non-overlapping intervals, as seen in "greater than" problems.
Special Cases
- If c is Negative: The rules change if the inequality is set to a negative number.
- For |ax + b| < -5, there is no solution. An absolute value can never be less than a negative number.
- For |ax + b| > -5, the solution is all real numbers. An absolute value is always greater than any negative number.
Frequently Asked Questions
Why does the inequality sign flip for one of the cases?
When you solve an inequality and have to multiply or divide both sides by a negative number, you must flip the direction of the inequality sign. In the second case of an absolute value inequality (e.g., ax + b < -c), if 'a' is negative, you will have to divide by a negative 'a' to isolate x, which requires flipping the sign.
How do I graph the solution on a number line?
You mark the endpoint(s) on a number line. Use an open circle (o) for endpoints that are not included (for < and >) and a closed circle (•) for endpoints that are included (for ≤ and ≥). Then, you shade the region of the number line that represents the solution set—either the bounded area between two points or the two unbounded areas going out to infinity.
How is this different from solving an absolute value equation?
An absolute value equation seeks the specific point(s) that are an exact distance from zero, resulting in one or two numerical answers. An inequality seeks all the points that are within or beyond a certain distance, resulting in a range or set of ranges as the solution.