A Budget Problem You Can Solve in Seconds
Maya is planning a birthday party for her daughter. She has $120 to spend on goody bags. Each bag costs $4 to assemble, and she already spent $18 on decorative boxes to hold them. How many bags can she make without going over budget? This is an inequality in disguise. The cost is 4x + 18, and it must stay at or below 120. Written out: 4x + 18 <= 120. Solve it and you get the maximum number of bags Maya can assemble. Inequalities show up everywhere once you start looking for them. Budgets, schedules, grade cutoffs, recipe scaling, and dosage limits are all inequality problems wearing different clothes.
What This Calculator Does and Why It Matters
This inequality solver handles linear inequalities of the form ax + b < c, where the comparison can be less than, greater than, less than or equal to, or greater than or equal to. You enter the coefficient a, the constant b, the comparison operator, and the right-side constant c. The tool returns the solution, the interval notation, a step-by-step breakdown, and a number line visualization.
Inequalities are the bridge between exact equations and real-world constraints. An equation finds one specific answer. An inequality finds a whole range of acceptable answers, which is usually what you actually need. If you also need to find a single exact value, our Linear Equation Solver handles equations of the same form. For second-degree problems, the Quadratic Equation Calculator solves ax squared plus bx plus c equals zero.
How the Calculation Works
A linear inequality looks like this:
ax + b < c
Solving it mirrors solving a linear equation, with one special rule. First, subtract b from both sides to isolate the x term. That gives ax < c minus b. Second, divide both sides by a to get x by itself. Here is the catch: if a is negative, you must flip the inequality sign. Dividing by a negative number reverses the relationship between the two sides. This is the single most common source of errors in inequality work.
Example: -2x + 5 < 11
Subtract 5: -2x < 6
Divide by -2 and flip: x > -3
The solution is x > -3. In interval notation that is (-3, inf), meaning every number greater than negative three. Interval notation uses parentheses for excluded endpoints and square brackets for included endpoints. An open parenthesis means the boundary value itself is not part of the solution. A square bracket means it is included.
x < 3 becomes (-inf, 3)
x <= 3 becomes (-inf, 3]
x > 3 becomes (3, inf)
x >= 3 becomes [3, inf)
Infinity always gets a parenthesis because it is not a real number you can reach or include.
How to Use the Calculator
- Enter the coefficient a. This is the number multiplying x. Use a negative number if the term subtracts.
- Enter the constant b. This is the number added or subtracted after the x term.
- Pick the comparison operator: less than, greater than, less than or equal to, or greater than or equal to.
- Enter the right-side constant c. This is the number on the other side of the inequality.
- Read the solution, interval notation, number line, and step-by-step working on the right.
Example Calculations
Example 1: Maya's Party Budget
Maya has $120 for goody bags. Each bag costs $4, and she spent $18 on boxes. The inequality is 4x + 18 <= 120. Subtract 18 from both sides to get 4x <= 102. Divide by 4 to get x <= 25.5. Since she cannot make half a bag, she can assemble at most 25 goody bags. The interval notation is (-inf, 25.5], and the number line shows a filled circle at 25.5 with shading to the left.
Example 2: Passing Grade Threshold
Jordan needs at least a 70 average across three tests to pass his math class. He scored 58 and 64 on the first two. Let x be his third score. The average is (58 + 64 + x) divided by 3, which must be at least 70. That simplifies to (122 + x) divided by 3 >= 70. Multiply both sides by 3 to get 122 + x >= 210. Subtract 122 to get x >= 88. Jordan needs an 88 or higher on the third test. The interval notation is [88, inf), with a filled circle at 88 and shading to the right.
Example 3: Negative Coefficient Flip
A small business owner, Priya, tracks her monthly loss. Her loss shrinks by $30 each month as she turns things around. After x months her total loss is 500 minus 30x dollars, and she wants it under $200. The inequality is 500 - 30x < 200. Subtract 500 to get -30x < -300. Divide by -30 and flip the sign to get x > 10. Priya needs more than 10 months to push her loss below $200. The interval notation is (10, inf) with an open circle at 10.
Real-World Scenarios
Budget Planning with Fixed Costs
A freelance designer charges $45 per hour and has $300 in monthly software subscriptions. She wants her monthly revenue to exceed $4,000. The inequality 45x + 300 > 4000 gives x > 82.2 hours. She needs to bill more than about 82 hours per month. This kind of calculation helps freelancers set realistic work targets and decide when to raise rates instead of working more hours.
Minimum Wage Hours to Cover Rent
A part-time worker earns $16 per hour after tax and pays $900 in rent. He wants at least $150 left over each month after rent. The inequality 16x - 900 >= 150 simplifies to 16x >= 1050, so x >= 65.6 hours. He needs to work at least 66 hours per month. Inequalities turn vague goals like "have some money left" into concrete numbers you can plan around.
Safe Temperature Ranges
A lab technician stores samples that must stay between 2 and 8 degrees Celsius. A refrigerator cools at a rate modeled by an inequality against time. If the cooling rate is -0.5 degrees per minute from a starting temperature, the technician uses inequalities to find the safe time window. Real constraints are almost always ranges, not single values, which is why inequalities are the right tool.
Common Mistakes to Avoid
- Forgetting to flip the sign: When you divide or multiply both sides by a negative number, the inequality reverses. This is the number one error in inequality algebra. Always check the sign of a before dividing.
- Confusing open and closed circles: An open circle means the endpoint is excluded, used for strict less than or greater than. A filled circle means the endpoint is included, used for less than or equal to and greater than or equal to. Mixing these up changes the meaning of your solution.
- Writing the interval backwards: The smaller number always goes on the left. Writing (5, -inf) is wrong. It should be (-inf, 5). Infinity is not a number, so it always takes a parenthesis, never a bracket.
- Treating inequalities like equations: An equation gives one answer. An inequality gives a range. If you stop after finding the boundary value and forget to state the direction, you have not fully solved the problem.
Limitations of This Calculator
This tool solves linear inequalities in one variable, of the form ax + b compared to c. It does not solve quadratic inequalities, systems of inequalities, or inequalities with two variables. It also does not handle compound inequalities with "and" or "or" connectors. For quadratic problems, use the Quadratic Equation Calculator. For single-value linear problems, the Linear Equation Solver is the right fit.
Authoritative Research and Resources
- Khan Academy: Solving Equations and Inequalities - Khan Academy offers a full algebra module covering linear inequalities with practice problems and video lessons. It is a solid starting point for students who want guided practice alongside a textbook.
- Purplemath: Solving Linear Inequalities - Purplemath provides clear written explanations of inequality solving, including the sign-flip rule and interval notation. It is widely referenced in high school and community college algebra courses.
- Paul's Online Math Notes: Linear Inequalities - Hosted by Lamar University, these notes walk through solving linear inequalities with worked examples. The site is a trusted reference for college algebra and calculus students.