Absolute Value Inequalities Calculator: Solve Inequalities with Absolute Values
Table of Contents - Absolute Value Inequalities
- How to Use This Calculator
- Understanding Absolute Value Inequalities
- How to Solve Absolute Value Inequalities Manually
- Real-World Applications
- Common Mistakes and How to Avoid Them
- Related Topics
- How This Calculator Works
- FAQs
How to Use This Calculator - Absolute Value Inequalities
Enter your absolute value inequality in the input field. Examples:
- |x| less than 5
- |x| greater than 3
- |2x - 1| less than or equal to 7
- |x + 3| greater than or equal to 4
Select the inequality type from the dropdown (less than, greater than, less than or equal to, greater than or equal to).
Click "Calculate" to see:
- The solution set (interval notation and inequality form)
- Step-by-step explanation
- Number line visualization
- Verification tips
The calculator handles both "less than" (bounded) and "greater than" (unbounded) absolute value inequalities.
Understanding Absolute Value Inequalities
If absolute value equations ask "where exactly?", then absolute value inequalities ask "where in a range?" They describe zones rather than specific points.
Think about temperature control again. |temperature - 20| less than 2 doesn't just give you specific acceptable temperatures—it gives you a whole range: between 18 and 22 degrees. That's what makes inequalities so powerful for real-world applications.
The fundamental split:
There are two completely different types of absolute value inequalities, and they behave in opposite ways:
Type 1: "Less than" inequalities (|x| less than a) These describe values NEAR zero—within a certain distance. The solution is an interval: -a less than x less than a.
Example: |x| less than 5 means x is between -5 and 5. Picture it: you can be anywhere from -5 to 5 on the number line.
Type 2: "Greater than" inequalities (|x| greater than a) These describe values FAR from zero—beyond a certain distance. The solution is two separate regions: x less than -a OR x greater than a.
Example: |x| greater than 5 means x is either less than -5 or greater than 5. Picture it: you're outside the interval from -5 to 5, in either direction.
The intuition:
"Less than" gives you one connected interval (everything close to zero). "Greater than" gives you two separate regions (everything far from zero).
This distinction is crucial. Mix them up and your solution will be completely wrong.
Visual thinking:
For |x| less than 5, imagine a circle of radius 5 centered at zero. You want everything inside. For |x| greater than 5, you want everything outside that circle.
Why the difference matters:
In quality control, |error| less than 0.5 means acceptable products (one range). In safety zones, |distance| greater than 10 means "stay away" (two regions on opposite sides).
How to Solve Absolute Value Inequalities Manually
The solution method depends entirely on whether you have "less than" or "greater than."
LESS THAN Inequalities: |expression| less than a
These become a compound inequality with AND.
Example 1: |x| less than 7
Step 1: Recognize this is "less than" type (one interval solution) Step 2: Write the compound inequality: -7 less than x less than 7 Step 3: Solution: x is in the interval (-7, 7)
Verification: Try x = 0: |0| = 0 less than 7 ✓ Try x = -5: |-5| = 5 less than 7 ✓ Try x = 10: |10| = 10, which is NOT less than 7 ✓ (correctly outside our solution)
Example 2: |x - 3| less than 5
Step 1: Set up compound inequality -5 less than x - 3 less than 5
Step 2: Solve by adding 3 to all parts -5 + 3 less than x less than 5 + 3 -2 less than x less than 8
Step 3: Solution in interval notation: (-2, 8)
Verification: Try x = 0: |0 - 3| = 3 less than 5 ✓ Try x = 5: |5 - 3| = 2 less than 5 ✓ Try x = 10: |10 - 3| = 7, which is NOT less than 5 ✓
Example 3: |2x + 1| less than or equal to 9
Step 1: Set up compound inequality -9 less than or equal to 2x + 1 less than or equal to 9
Step 2: Subtract 1 from all parts -10 less than or equal to 2x less than or equal to 8
Step 3: Divide all parts by 2 -5 less than or equal to x less than or equal to 4
Step 4: Solution: [-5, 4] (brackets because we include endpoints)
GREATER THAN Inequalities: |expression| greater than a
These split into two separate inequalities with OR.
Example 4: |x| greater than 3
Step 1: Recognize this is "greater than" type (two separate regions) Step 2: Split into two cases:
- Case 1: x greater than 3 (right side)
- Case 2: x less than -3 (left side)
Step 3: Solution: x less than -3 OR x greater than 3 In interval notation: (-∞, -3) ∪ (3, ∞)
Verification: Try x = 5: |5| = 5 greater than 3 ✓ Try x = -10: |-10| = 10 greater than 3 ✓ Try x = 0: |0| = 0, which is NOT greater than 3 ✓
Example 5: |x + 2| greater than 4
Step 1: Split into two cases
- Case 1: x + 2 greater than 4
- Case 2: x + 2 less than -4
Step 2: Solve Case 1 x + 2 greater than 4 x greater than 2
Step 3: Solve Case 2 x + 2 less than -4 x less than -6
Step 4: Combine with OR: x less than -6 OR x greater than 2 In interval notation: (-∞, -6) ∪ (2, ∞)
Example 6: |3x - 6| greater than or equal to 12
Step 1: Split into cases
- Case 1: 3x - 6 greater than or equal to 12
- Case 2: 3x - 6 less than or equal to -12
Step 2: Solve Case 1 3x - 6 greater than or equal to 12 3x greater than or equal to 18 x greater than or equal to 6
Step 3: Solve Case 2 3x - 6 less than or equal to -12 3x less than or equal to -6 x less than or equal to -2
Step 4: Solution: x less than or equal to -2 OR x greater than or equal to 6 In interval notation: (-∞, -2] ∪ [6, ∞)
Memory trick:
"Less than" = sandwich (squeeze between two values) "Greater than" = split (break into two pieces)
Real-World Applications
Manufacturing tolerances: A bolt must be 5 cm long with tolerance of 0.1 cm. Acceptable lengths satisfy |length - 5| less than or equal to 0.1, giving 4.9 to 5.1 cm.
Temperature safety: A chemical reaction requires temperature between 80°C and 100°C. This is |temp - 90| less than 10, where 90 is the midpoint.
Speed limits: A highway has a minimum speed of 45 mph and maximum of 65 mph. The speed range can be written as |speed - 55| less than or equal to 10.
Quality control in food: Cereal boxes labeled 500g must contain 500g ± 5g. Acceptable weights satisfy |weight - 500| less than or equal to 5, meaning 495g to 505g.
Medical dosing safety: A medication is safe between 80mg and 120mg. This is |dose - 100| less than or equal to 20.
Audio levels: Sound should be louder than 60 dB to be heard clearly but quieter than 90 dB to prevent damage. However, this is better expressed as two separate inequalities rather than one absolute value inequality, since there's no natural center point.
Sensor accuracy: A sensor reading should be within 2% of the true value. If the true value is 100, acceptable readings satisfy |reading - 100| less than or equal to 2.
Time windows: A train should arrive within 5 minutes of scheduled time. If scheduled for 3:00 PM (180 minutes after noon), acceptable arrival satisfies |arrival - 180| less than or equal to 5.
Distance constraints: A drone must stay more than 50 meters from the airport. This is |distance - airport| greater than 50, describing two regions (one on each side of the airport).
Common Mistakes and How to Avoid Them
Mistake 1: Mixing up "less than" and "greater than" logic
Wrong: Solving |x| less than 5 as "x less than -5 OR x greater than 5"
Right: |x| less than 5 becomes -5 less than x less than 5 (one interval)
Why it matters: You'll get the exact opposite of the correct answer. "Less than" gives one interval, "greater than" gives two regions.
Mistake 2: Forgetting to flip the inequality when multiplying/dividing by negatives
Wrong: Solving -2x greater than 6 as x greater than -3
Right: -2x greater than 6 becomes x less than -3 (inequality flips)
Why it happens: This is a basic algebra rule that's easy to forget when focused on the absolute value part.
Mistake 3: Using AND instead of OR for "greater than" inequalities
Wrong: Writing |x| greater than 5 as "x greater than 5 AND x less than -5"
Right: It's "x greater than 5 OR x less than -5"
Why it matters: AND means both must be true simultaneously (impossible here). OR means either can be true.
Mistake 4: Including the boundary when you shouldn't
Wrong: Solving |x| less than 5 as -5 less than or equal to x less than or equal to 5
Right: For strict inequality (less than, not less than or equal to), use open interval: -5 less than x less than 5
Notation: Parentheses ( ) mean open (not included), brackets [ ] mean closed (included).
Mistake 5: Not solving for x completely
Wrong: Stopping at -7 less than 2x less than 7
Right: Divide all parts by 2 to get -3.5 less than x less than 3.5
Why it happens: We get excited about setting up the compound inequality and forget to finish solving.
Mistake 6: Solving absolute value of zero incorrectly
Wrong: Thinking |x - 3| less than 0 has solutions
Right: Absolute value is never negative, so no solutions exist.
Special case: |x - 3| less than or equal to 0 only has solution x = 3 (where the absolute value equals exactly zero).
Recovery strategies:
Always draw a number line to visualize your answer. Does it make sense? Test a few values to verify they satisfy the original inequality. Remember the fundamental rule: "less than" squeezes, "greater than" splits.
Related Topics
Absolute Value Equations: When you need exact values instead of ranges, you solve |x| = a. See our Absolute Value Equation Calculator.
Compound Inequalities: These combine two inequalities with AND or OR. Absolute value inequalities are a special type of compound inequality.
Interval Notation: A compact way to write solution sets. (-3, 5) means all numbers between -3 and 5, not including endpoints.
Linear Inequalities: The simpler cousin without absolute values. Once you split an absolute value inequality, you solve linear inequalities.
Distance and Tolerance: The applications of absolute value inequalities connect deeply to measurement and error analysis.
How This Calculator Works
The calculator uses different logic for "less than" vs "greater than" inequalities:
For |expression| less than a:
Step 1: Verify a is positive (if not, special case)
Step 2: Create compound inequality:
-a less than expression less than a
Step 3: Solve the compound inequality for the variable
Step 4: Express as interval notation
For |expression| greater than a:
Step 1: Verify a is positive
Step 2: Split into two inequalities:
expression greater than a OR
expression less than -a
Step 3: Solve each inequality separately
Step 4: Combine with union symbol
Special cases:
If a less than 0:
For "less than": no solution
For "greater than": all real numbers
If a = 0:
For "less than": no solution
For "greater than": all numbers except solution to expression = 0
All work is done in your browser. Nothing is transmitted to servers.
FAQs
What's the difference between |x| less than 5 and |x| greater than 5?
The first gives you everything between -5 and 5 (one interval). The second gives you everything less than -5 or greater than 5 (two separate regions). They're complete opposites.
Why do "less than" inequalities use AND while "greater than" use OR?
For "less than," both conditions must hold: x must be greater than the negative value AND less than the positive value. For "greater than," only one condition must hold: x can be less than the negative value OR greater than the positive value.
How do I know if I should include the endpoints?
Look at the inequality symbol. "Less than" (less than) and "greater than" (greater than) exclude endpoints—use parentheses ( ). "Less than or equal to" (less than or equal to) and "greater than or equal to" (greater than or equal to) include endpoints—use brackets [ ].
Can an absolute value inequality have no solution?
Yes. |x| less than -3 has no solution because absolute value is never negative. Also, compound inequalities can sometimes create contradictions.
Can an absolute value inequality have all real numbers as solutions?
Yes. |x| greater than -5 is true for all real numbers because absolute value is always non-negative, so it's always greater than -5.
What does the union symbol mean?
The union symbol ∪ in (-∞, -3) ∪ (5, ∞) means "or"—x is in the first interval OR the second interval. It combines the two solution regions.
How do I graph these on a number line?
For "less than," shade the region between the two boundary points. For "greater than," shade two regions extending outward from the boundaries. Use open circles for strict inequalities, closed circles when endpoints are included.
What if I have |something| less than or equal to 0?
Since absolute value is never negative, this is only true when the absolute value equals exactly zero. Solve expression = 0 for that single point.
What if both sides have absolute values?
You need to consider multiple cases based on the signs of both expressions. It gets complex—consider graphing or testing critical points.
How do I handle negative coefficients inside the absolute value?
Factor them out. For |-2x + 6| less than 8, rewrite as |-(2x - 6)| = |2x - 6| less than 8, since |-a| = |a|.
Why can't I just remove the absolute value bars?
Because absolute value affects negative and positive values differently. You must account for both possibilities, which is why we create cases or compound inequalities.
Can I square both sides to eliminate absolute value?
For inequalities, squaring can introduce errors because squaring doesn't preserve the inequality direction for negative numbers. Stick with the case method.
What's the difference between (-3, 5) and [-3, 5]?
Parentheses mean open interval (endpoints NOT included). Brackets mean closed interval (endpoints included). (-3, 5] mixes both: -3 not included, 5 included.
How do I verify my solution?
Test a value from each region. For |x| less than 5 giving (-5, 5), test x = 0 (should work) and x = 10 (should not work).
What if I get a contradiction?
Like -3 less than x less than -5? This means no values satisfy both conditions—no solution exists.
What if the inequality simplifies to always true?
Like 5 less than 10? Then all real numbers are solutions, written as (-∞, ∞) or ℝ.
How do these relate to distance?
|x - a| less than b means "distance from a is less than b"—you're within b units of a. |x - a| greater than b means "distance from a is greater than b"—you're more than b units from a.
Can I have variables on both sides?
Yes, but it's more complex. You'll need to consider cases based on which expression is larger, then solve each case.
What's the fastest way to solve these?
Memorize the patterns: "less than" gives -a less than expression less than a, "greater than" gives expression less than -a or expression greater than a. Then solve normally.
Why are these used in real life?
Tolerances, safety ranges, acceptable variations—anytime you need to specify "close enough" or "too far away," you use absolute value inequalities.
Additional Notes
Absolute value inequalities are essential for describing ranges and tolerances in science, engineering, and daily life. The key is recognizing the fundamental difference between "less than" (bounded range) and "greater than" (unbounded regions).
Practice visualizing solutions on a number line. This builds intuition and helps you catch errors. When in doubt, test a few values to verify your solution makes sense.
Master this concept and you'll be able to translate real-world constraints into mathematical language and vice versa—a skill that's valuable far beyond the classroom.