Inequality Solver — Solve ax + b > c
Solution
--
Steps
--
Interval Notation
--
How Inequality Solving Works
A linear inequality is a mathematical statement comparing a linear expression to a value using inequality symbols: greater than (>), less than (<), greater than or equal to (>=), or less than or equal to (<=). Unlike equations that have one solution, inequalities typically have infinitely many solutions forming a continuous range on the number line. According to the Khan Academy algebra curriculum, linear inequalities are among the most fundamental concepts in algebra, forming the basis for optimization, constraint modeling, and linear programming.
This calculator solves inequalities of the form ax + b > c (or <, >=, <=) by isolating x. The process mirrors solving linear equations with one critical difference: multiplying or dividing both sides by a negative number reverses the inequality direction. According to the Mathematical Association guidelines, forgetting to flip the sign when dividing by a negative is the single most common error students make with inequalities, accounting for approximately 40% of mistakes on standardized tests involving this topic.
How Linear Inequalities Are Solved
The general form of a linear inequality is ax + b > c, where a, b, and c are known constants and x is the variable. The solution process:
- Step 1: Subtract b from both sides: ax > c - b
- Step 2: Divide both sides by a: x > (c - b) / a
- Critical rule: If a < 0, flip the inequality symbol when dividing
Worked example: Solve 3x + 5 > 20. Step 1: Subtract 5 from both sides: 3x > 15. Step 2: Divide by 3 (positive, so no flip): x > 5. In interval notation: (5, +infinity). Every number greater than 5 satisfies this inequality.
Negative coefficient example: Solve -2x + 8 >= 14. Step 1: Subtract 8: -2x >= 6. Step 2: Divide by -2 (negative, so flip >= to <=): x <= -3. In interval notation: (-infinity, -3]. The sign flips because multiplying by a negative reverses the number line ordering.
Key Terms You Should Know
- Inequality -- A mathematical statement that two expressions are not necessarily equal, using the symbols >, <, >=, or <=. Unlike equations, inequalities have a range of solutions rather than a single point.
- Interval Notation -- A compact way to write solution ranges: (a, b) excludes both endpoints, [a, b] includes both. Parentheses always appear next to infinity symbols since infinity cannot be reached.
- Strict Inequality -- Uses > or < without the equal part. The boundary value is excluded from the solution set (open circle on number line, parenthesis in interval notation).
- Non-strict (Weak) Inequality -- Uses >= or <=, including the boundary value in the solution set (filled circle on number line, bracket in interval notation).
- Solution Set -- The set of all values that make the inequality true. For linear inequalities with nonzero coefficient, this is always a half-line (ray) on the number line.
- Empty Set -- The result when no value satisfies the inequality (e.g., 0x + 3 > 10 simplifies to 3 > 10, which is always false).
Common Inequality Types and Solution Patterns
Understanding solution patterns helps students quickly verify their work. The following table summarizes common inequality scenarios:
| Inequality Type | Example | Solution | Interval Notation |
|---|---|---|---|
| Positive coefficient, > | 2x + 1 > 7 | x > 3 | (3, +inf) |
| Positive coefficient, <= | 4x - 3 <= 13 | x <= 4 | (-inf, 4] |
| Negative coefficient (flips) | -3x + 6 > 0 | x < 2 | (-inf, 2) |
| Zero coefficient (all reals) | 0x + 3 < 10 | All real numbers | (-inf, +inf) |
| Zero coefficient (no solution) | 0x + 7 < 2 | No solution | Empty set |
| Fractional result | 5x + 2 >= 9 | x >= 1.4 | [7/5, +inf) |
Practical Examples
Example 1 -- Budget constraint: You have $200 for groceries and each bag costs $35. How many bags can you buy? The inequality is 35x <= 200. Dividing: x <= 5.71. Since you cannot buy a fraction of a bag, you can buy at most 5 bags. This demonstrates how inequalities model real-world constraints.
Example 2 -- Temperature conversion: A chemical reaction requires temperatures above 100°C. If the relationship is F = 1.8C + 32, what Fahrenheit temperatures work? Solve 1.8C + 32 > 100: 1.8C > 68, so C > 37.78. Converted back: F > 212°F. Use our temperature converter to verify conversions.
Example 3 -- Grading threshold: A student needs an average of at least 70 across 4 tests to pass. They scored 65, 72, and 68. What minimum score on test 4 is needed? The inequality is (65 + 72 + 68 + x) / 4 >= 70, which simplifies to 205 + x >= 280, giving x >= 75. Use our grade calculator for multi-test averages or our equation solver for more complex expressions.
Tips and Strategies for Solving Inequalities
- Always check the sign of the coefficient before dividing: If a is negative, the inequality flips. Write a reminder note ("FLIP!") next to the division step until it becomes automatic.
- Verify your answer with a test point: After solving, pick a number in your solution range and substitute it back into the original inequality. If it works, your solution is likely correct.
- Watch for the zero coefficient case: If a = 0, the x term vanishes and you are left comparing constants. The answer is either "all real numbers" or "no solution."
- Convert to interval notation carefully: Greater-than uses parentheses at the boundary and extends right toward +infinity. Less-than extends left toward -infinity. Include the boundary (bracket) only for >= or <=.
- Draw a number line: Visualizing the solution on a number line helps catch errors. Shade the solution region and use open/filled circles to mark strict/non-strict boundaries.
- Practice with negative coefficients: Since the sign-flip rule causes the most errors, deliberately practice problems with negative coefficients until the rule becomes second nature.
Frequently Asked Questions
Why does the inequality sign flip when multiplying or dividing by a negative number?
Multiplying or dividing both sides of an inequality by a negative number reverses the ordering of all numbers on the number line. For example, 2 < 3 is true, but multiplying both sides by -1 gives -2 and -3, and since -2 > -3, the inequality must flip to remain true. This happens because negation mirrors numbers across zero: positive numbers become negative and vice versa, reversing their relative positions. This rule applies to both multiplication and division by any negative value. Forgetting to flip the sign is the single most common mistake students make when solving inequalities, according to math education research.
What is interval notation and how do I read it?
Interval notation is a compact way to describe a continuous range of numbers using parentheses and brackets. A parenthesis ( or ) means the endpoint is excluded (open), while a bracket [ or ] means the endpoint is included (closed). For example, (3, 7) means all numbers between 3 and 7, not including 3 or 7. [3, 7] includes both endpoints. (-infinity, 5) means all numbers less than 5. [2, +infinity) means all numbers greater than or equal to 2. Infinity always uses a parenthesis because infinity is a concept, not a number that can be reached or included. The empty set (no solution) is written as the symbol with a line through the zero.
Can a linear inequality have no solution or infinitely many solutions?
Yes to both. When the coefficient of x is zero (a = 0), the inequality reduces to a comparison of constants. If the resulting statement is false (like 5 > 10), there is no solution -- no value of x can make it true, and the solution set is the empty set. If the resulting statement is true (like 5 < 10), then every real number satisfies the inequality, giving infinitely many solutions expressed as (-infinity, +infinity). For standard linear inequalities where a is not zero, the solution is always a half-line: either all numbers above a value or all numbers below a value.
What is the difference between strict and non-strict inequalities?
Strict inequalities use the symbols less-than and greater-than (without the equal bar) and exclude the boundary point from the solution. For example, x > 5 means x can be 5.001 or 6 or 100, but not exactly 5. Non-strict (or weak) inequalities use the symbols with an equal bar and include the boundary point. For example, x >= 5 means x can be exactly 5, or 6, or 100. In interval notation, strict uses parentheses at the boundary (5, infinity) while non-strict uses brackets [5, infinity). On a number line, strict inequalities show an open circle at the boundary point, while non-strict show a filled circle.
How do I solve compound inequalities like 2 < 3x + 1 < 10?
A compound inequality like 2 < 3x + 1 < 10 means both conditions must be true simultaneously. Solve by treating it as two separate inequalities joined by AND. Step 1: subtract 1 from all three parts: 1 < 3x < 9. Step 2: divide all three parts by 3: 1/3 < x < 3. The solution is the intersection of both conditions, written in interval notation as (1/3, 3). For compound inequalities joined by OR (like x < -2 or x > 5), the solution is the union of the two intervals: (-infinity, -2) union (5, infinity). This calculator handles single linear inequalities; for compound inequalities, solve each part separately and combine.
Where are linear inequalities used in real life?
Linear inequalities appear throughout everyday life and professional fields. In budgeting, if you earn $3,000 per month and rent is $1,200, you need expenses to satisfy x + 1200 <= 3000, meaning other spending must be at most $1,800. In manufacturing, if a machine can produce at most 500 units per day, the production constraint is x <= 500. Engineers use inequalities for safety margins, such as a bridge load limit of x <= 10 tons. In linear programming, businesses optimize profit or cost subject to multiple inequality constraints on resources, labor, and demand. Standardized tests including the SAT, GRE, and ACT regularly test inequality-solving skills.