Inequality Calculator

An inequality calculator takes a statement like 2x − 4 > 10 and tells you exactly which values of x make it true. The answer is not a single number but a whole range - in this case x > 7, or in interval notation (7, +∞). This tool isolates x the same way you would by hand, shows every step, and - crucially - flips the inequality sign automatically whenever you divide by a negative number, the one rule that trips up almost everyone.

A worked example

Take 2x − 4 > 10. First add 4 to both sides to get 2x > 14. Then divide both sides by 2 (a positive number, so the sign stays) to get x > 7. Check it: plug in x = 8 → 2(8) − 4 = 12 > 10 ✓, and x = 7 → 10 > 10 is false, so 7 itself is excluded (an open circle on the number line). That exclusion is why the interval uses a parenthesis: (7, +∞).

The formula and the steps

Every linear inequality this tool handles fits the pattern:

The solving recipe is three moves:

1. Subtract b from both sides: a·x (op) c − b.
2. Divide by a: x (op) (c − b) ÷ a.
3. Flip the sign if a < 0: dividing by a negative number reverses the direction of the inequality.

Why you flip the sign

This is the rule that separates inequalities from equations. Consider a true statement like 3 < 5. Multiply both sides by −1 and you get −3 and −5. Is −3 < −5? No - −3 is the larger number, so the correct statement is −3 > −5. The direction had to reverse. The same thing happens any time you multiply or divide both sides by a negative value, which is why this calculator highlights the flip whenever your coefficient a is negative.

How to use this calculator

Enter the four parts of your inequality and press Solve:

1. a - the coefficient multiplying x (use a negative value to see the sign flip).
2. b - the constant added to the x term (enter a negative number if it is subtracted).
3. Comparison - choose <, ≤, > or ≥.
4. c - the value on the right-hand side.

The live preview shows your inequality as you type. After you solve, read the answer in both inequality and interval notation, see the boundary value and whether it is included, and follow the step table and number line to understand the working.

Reading interval notation

Interval notation is a compact way to write a solution range:

• Parenthesis ( ) = endpoint excluded, used for strict < and >.
• Square bracket [ ] = endpoint included, used for ≤ and ≥.
• −∞ and +∞ always take a parenthesis, because infinity is never a reachable endpoint.

So x > 7 becomes (7, +∞), while x ≤ 7 becomes (−∞, 7].

Who this is for

• Algebra students learning to isolate a variable and remembering when to flip the sign.
• Parents and tutors who want to show each step rather than just the answer.
• Anyone checking homework who needs the result in both inequality and interval form.
• Returning learners brushing up before a placement test or a stats/economics course that uses inequalities.

Three quick scenarios

• Positive coefficient: 3x + 2 ≤ 11 → 3x ≤ 9 → x ≤ 3, interval (−∞, 3], closed dot at 3.
• Negative coefficient (sign flips): −2x + 1 < 7 → −2x < 6 → divide by −2 and flip → x > −3, interval (−3, +∞).
• No x term: 0x + 5 > 2 reduces to 5 > 2, which is always true, so the answer is all real numbers.

Key terms

• Coefficient (a): the number multiplying x. Its sign decides whether the inequality flips.
• Boundary / critical value: the number where both sides are equal - the dividing line between solutions and non-solutions.
• Strict vs. non-strict: < and > are strict (boundary excluded); ≤ and ≥ include the boundary.
• Solution set: all the values of x that make the statement true - a ray on the number line.

What changes the result

• Sign of a: a negative coefficient reverses the inequality direction.
• Strict vs. inclusive operator: changes only whether the boundary point is part of the solution (open vs. closed dot).
• Values of b and c: shift the boundary value left or right but do not change the direction.
• a = 0: removes x entirely, giving either no solution or all real numbers.

Tips for solving by hand

• Undo in reverse order: handle addition/subtraction first, then multiplication/division - the opposite of the order of operations.
• Watch the sign every time you divide: circle the operator and flip it the moment you divide by a negative.
• Test a point: substitute one number from your answer back into the original inequality to confirm it.
• Keep the boundary, change only its dot: the number is the same for strict and non-strict; only inclusion differs.

Where inequalities show up in real life

Inequalities are not just textbook exercises - they describe any situation with a limit, a threshold, or a "at least / at most" condition. A few everyday examples translate directly into the a·x + b (op) c form this calculator solves:

• Budgeting: "I have $200 and each ticket costs $15 plus a $20 fixed fee - how many can I buy?" becomes 15x + 20 ≤ 200, giving x ≤ 12, so at most 12 tickets.
• Grades: "I need an average of at least 90 over four tests and already have 88, 92, 85" sets up an inequality for the score still needed on the last test.
• Business break-even: a firm turns a profit only when revenue beats cost, which is an inequality in units sold.
• Speed and distance: "to arrive within 2 hours at speed x" produces a lower bound on x.

Once you write the situation as a single linear inequality, the solving steps are exactly the ones above. If your problem is about a percentage of a budget rather than a fixed threshold, the Percentage Calculator can help you find the numbers to plug in first.

A worked word problem

Suppose a phone plan charges a $25 base fee plus $0.10 per minute, and you want to keep the bill under $40. Let x be the number of minutes. The bill is 0.10x + 25, and "under $40" means 0.10x + 25 < 40. Subtract 25 from both sides to get 0.10x < 15, then divide by 0.10 (a positive number, so the sign stays) to get x < 150. So you can talk for up to - but not including - 150 minutes and stay under budget. In interval notation that is (−∞, 150), with an open dot at 150 because spending exactly $40 would not be "under" $40. Enter a = 0.10, b = 25, the < operator, and c = 40 to see the same result with every step laid out.

When x appears on both sides

This calculator expects the inequality already arranged as a·x + b (op) c, with all the x terms collected into a single coefficient a. If your problem has x on both sides, do one tidy-up step first. Take 5x + 3 > 2x + 12: subtract 2x from both sides to get 3x + 3 > 12, which is now in standard form with a = 3, b = 3, and c = 12. From there the calculator finishes it: 3x > 9, so x > 3. The rule of thumb is to move every x term to the side where the coefficient stays positive when possible - that way you avoid an extra sign flip. The same trick works for constants: gather all the plain numbers on the opposite side from x.

Graphing the solution on a number line

The number line is the picture of your answer. After solving, you draw a single point at the boundary value and then shade a ray in the direction the inequality points. The dot style encodes strictness: an open (hollow) circle for < or > because the boundary is not part of the solution, and a closed (filled) circle for ≤ or ≥ because it is. The direction of shading matches the variable: x > 3 shades to the right of 3 (larger numbers), while x < 3 shades to the left. A common slip is to read the original operator instead of the final one - after a sign flip the shading direction follows the solved inequality, not the problem you started with. This calculator draws the line for you so you can sanity-check your hand drawing against it.

Limitations

This is a single-inequality, single-variable solver. It does not handle compound inequalities such as 1 < 2x + 3 ≤ 7, inequalities with an x² term, absolute-value inequalities, or systems with more than one variable. It also assumes the inequality is already arranged as a·x + b (op) c; if x appears on both sides, move all the x terms to one side first and combine them into a single a. The output is a planning/learning aid, not a substitute for showing your own work on graded assignments.

How it compares to related tools

If your problem is not a plain linear inequality, a sister calculator fits better:

• For an x² equation, solve the roots with the Quadratic Formula Calculator, then test intervals.
• To find the slope and equation of a line from two points, use the Slope Calculator.
• For percentages and "what is X% of Y" problems, use the Percentage Calculator.
• For powers and roots, use the Exponent Calculator or the Square Root Calculator; for everything else, the Scientific Calculator.