Inequality Calculator
Solve inequality problems step-by-step with our free calculator. See formulas, worked examples, and clear explanations.
Reviewed for accuracy by Manoj Kumar, Mathematics Educator
Inequality Calculator
Calculator
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Formula: ax + b < cx + d (solve for x)
Worked example — x > -4.5 or (-4.5, Infinity)
Formula
ax + b < cx + d (solve for x)
Enter the coefficients and constants for both sides of the inequality. The calculator moves all variable terms to one side and constants to the other, then divides by the coefficient. If dividing by a negative number, the inequality sign is automatically flipped.
Worked Examples
Example 1: Basic Linear Inequality
Problem:Solve the inequality: 3x - 7 < 5x + 2
Solution:Move all x terms to left side: 3x - 5x < 2 + 7 Simplify: -2x < 9 Divide by -2 (flip the sign): x > -4.5 Interval notation: (-4.5, Infinity) Test: x = 0 gives 3(0) - 7 = -7 < 5(0) + 2 = 2, which is true.
Result:x > -4.5 or (-4.5, Infinity)
Example 2: Inequality with Equal Coefficients
Problem:Solve: 4x + 3 <= 4x + 8
Solution:Subtract 4x from both sides: 3 <= 8 This is always true. The x terms cancel out, leaving a true numerical statement. Therefore the solution is all real numbers. Interval notation: (-Infinity, Infinity)
Result:All real numbers: (-Infinity, Infinity)
Frequently Asked Questions
What is an inequality and how does it differ from an equation?
An inequality is a mathematical statement that compares two expressions using inequality symbols such as less than, greater than, less than or equal to, or greater than or equal to. Unlike equations which have specific solutions, inequalities typically have a range of solutions. For example, x > 3 means any value greater than 3 satisfies the inequality, giving infinitely many solutions. Equations use the equals sign and find exact values, while inequalities define regions on the number line. Understanding this distinction is crucial because many real-world problems involve constraints and ranges rather than exact values.
Why does the inequality sign flip when multiplying or dividing by a negative number?
The inequality sign flips because multiplying or dividing by a negative number reverses the order of numbers on the number line. Consider the true statement 3 < 5. If you multiply both sides by -1, you get -3 and -5. Since -3 is to the right of -5 on the number line, -3 > -5, so the inequality must flip. This is a fundamental rule that students frequently forget, leading to incorrect solutions. For example, solving -2x > 6 requires dividing by -2 AND flipping the sign, giving x < -3. Always check your answer by substituting a test point from your solution set back into the original inequality.
What is interval notation and how do I read it?
Interval notation is a compact way to describe a set of numbers between two endpoints. Parentheses ( ) mean the endpoint is NOT included (open interval), while brackets [ ] mean the endpoint IS included (closed interval). For example, (2, 5) means all numbers between 2 and 5, not including 2 or 5. [2, 5] includes both endpoints. (-Infinity, 3) means all numbers less than 3. [3, Infinity) means all numbers greater than or equal to 3. Infinity always uses parentheses because you cannot actually reach infinity. This notation is preferred in higher mathematics because it is concise and unambiguous compared to inequality notation.
How do you solve compound inequalities?
Compound inequalities combine two inequalities using AND or OR. An AND compound inequality like 2 < x + 1 < 7 requires both conditions to be true simultaneously, and you solve by performing the same operation on all three parts: subtracting 1 gives 1 < x < 6. An OR compound inequality like x < -2 OR x > 5 is satisfied when either condition is true. The solution set for AND is the intersection of the two individual solutions, while OR gives the union. Compound inequalities are common in real-world constraints, such as temperature ranges, acceptable measurements, and quality control specifications.
How are inequalities used in real-world applications?
Inequalities are essential for modeling constraints and optimization problems in real life. Budget constraints use inequalities to represent spending limits. Engineers use inequalities to define safety margins and tolerances. In business, profit inequalities help determine break-even points and minimum sales targets. Nutritionists use inequalities to plan diets that meet minimum nutrient requirements while staying within calorie limits. Speed limits, age restrictions, weight capacities, and temperature ranges are all everyday examples of inequalities. Linear programming, which optimizes business decisions, is entirely based on systems of linear inequalities.
What is the difference between strict and non-strict inequalities?
Strict inequalities use < (less than) or > (greater than) and do NOT include the boundary value. Non-strict (or weak) inequalities use <= (less than or equal to) or >= (greater than or equal to) and DO include the boundary value. This difference matters practically. If a bridge has a weight limit of 5 tons (weight <= 5 tons), a 5-ton truck can cross. If the limit were strictly less than 5 tons (weight < 5 tons), that same truck could not. In interval notation, strict inequalities use parentheses and non-strict use brackets at the boundary point. On a number line graph, strict inequalities use open circles and non-strict use filled circles.
How do you graph linear inequalities on a number line?
To graph a linear inequality on a number line, first solve for the variable to find the critical point. Place either an open circle (for strict inequalities < or >) or a closed/filled circle (for non-strict inequalities <= or >=) at the critical point. Then shade the appropriate direction: shade left for less than and shade right for greater than. For example, x > 3 gets an open circle at 3 with shading to the right. To verify your graph, pick a test point from the shaded region and substitute it into the original inequality. If it satisfies the inequality, your graph is correct. This visual representation helps build intuition about solution sets.
What is set-builder notation for inequalities?
Set-builder notation describes a set by stating the property that its members must satisfy. It uses the format {x | condition}, read as 'the set of all x such that condition is true.' For example, {x | x > 5} means the set of all real numbers greater than 5. This is equivalent to the interval (5, Infinity) and the inequality x > 5. Set-builder notation is more flexible than interval notation because it can describe complex conditions like {x | x is an integer and 0 < x < 10}. It is commonly used in formal mathematics, discrete math, and computer science where precise set definitions are needed.
How do absolute value inequalities work?
Absolute value inequalities involve expressions like |ax + b| < c or |ax + b| > c. For less-than types, |expression| < c means -c < expression < c, creating a compound AND inequality that produces a bounded interval. For greater-than types, |expression| > c means expression < -c OR expression > c, creating a compound OR inequality with two unbounded rays. For example, |x - 3| < 5 becomes -5 < x - 3 < 5, which simplifies to -2 < x < 8. And |2x + 1| > 7 becomes 2x + 1 < -7 OR 2x + 1 > 7, giving x < -4 OR x > 3. These appear frequently in error bounds and tolerance specifications.
What common mistakes should I avoid when solving inequalities?
The most common mistake is forgetting to flip the inequality sign when multiplying or dividing by a negative number. Another frequent error is incorrectly combining terms when moving them across the inequality sign. Students also sometimes treat inequalities like equations and write single-value answers instead of ranges. Confusion between strict and non-strict inequalities leads to incorrect interval notation. When solving compound inequalities, students often apply operations incorrectly to only part of the expression. Always verify your solution by substituting test points from both inside and outside the solution set. Checking boundary values is especially important for non-strict inequalities.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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