Solve inequalities with fractions instantly — see the LCD clearing step, full algebraic steps, number line graphs, and interval notation. Free, no sign-up required.
Why this page is built for fraction inequality intent
Fraction Inequality Solver
Built for inequalities with fraction coefficients and constants — shows the LCD clearing step, full algebraic isolation, sign-flip detection, and number line output.
Supported Input Styles
x/2+1>3 clears the fraction by multiplying both sides by 2 before solving.(3/4)x-2<=5 multiplies through by 4 to eliminate the fraction coefficient.x/3+x/2<4 finds the LCD of 3 and 2 (which is 6) and multiplies through.-(x/4)+3>=1 multiplies by 4 — the negative sign stays with x, not the denominator.x/2-1>x/3+2 has fraction terms on both sides — LCD clears all fractions at once.Math Keyboard
Tap symbols, numbers, or actions for fast linear-inequality input.
Result
The fraction inequality clears the LCD first, then isolates x > 4.
Step 1
Step 1 — Identify the fractions and find the LCD
Find the least common denominator of all fraction denominators in the inequality.
Step 2
Step 2 — Multiply both sides by the LCD
Multiply every term on both sides by the LCD to clear all fractions. If the LCD is positive, the symbol stays the same.
Before
x/2 + 1 > 3
After
x + 2 > 6
Step 3
Step 3 — Solve the resulting inequality
The inequality is now fraction-free. Isolate x using standard steps.
Before
x + 2 > 6
After
x > 4
Recent History
Saved locally in this browser so you can revisit recent fraction inequalities and compare how each LCD step was handled.
Calculator Types
Solve one-variable linear inequalities with steps, interval notation, and a clean number-line graph.
Solve two-step inequalities with the sign-flip rule, number line, and interval notation.
Solve multi-step inequalities with distribution, combining like terms, variable collection, and sign-flip detection.
Solve inequalities with fraction coefficients using the LCD clearing method, with steps, number line, and interval notation.
Solve quadratic inequalities with sign analysis, roots, interval notation, and a number-line graph.
Solve absolute value inequalities with case splitting, interval notation, and step-by-step explanations.
Solve compound inequalities with interval intersection, union logic, and graph output.
Solve rational inequalities with steps, excluded values, sign charts, and interval notation.
Zone 4
The fastest way to use this calculator is to type the inequality exactly as it appears — x/2 + 1 > 3, (3/4)x - 2 <= 5, or x/3 + x/2 < 4. Use the slash symbol / for fractions and standard keyboard symbols for the inequality sign. The parser identifies all fraction terms automatically and finds the LCD before beginning the solving steps.
After you enter a problem, the Steps tab shows the LCD clearing step first — the multiplication that removes all fractions from the inequality at once. This is the move that most students either skip or apply incorrectly, so the calculator shows it as a distinct step with the Before and After expressions side by side. The remaining steps follow the standard linear inequality pattern: combine like terms, collect variable terms, and isolate x.
If you are checking homework, compare the LCD clearing step to your own work first — that is where most fraction inequality errors originate. If you are learning the method for the first time, pay attention to whether the LCD is positive or negative. Multiplying by a positive LCD keeps the symbol the same. Multiplying by a negative LCD flips the symbol, though this situation is rare in standard fraction inequalities.
If the denominator contains a variable, move to the rational inequality calculator. If the fractions clear and the rest of the algebra becomes longer, the multi-step inequality calculator follows the same post-LCD cleanup pattern.
Enter an inequality with fractions such as x/2 + 1 > 3 or (3/4)x - 2 <= 5.
Review the LCD clearing step first, then follow the remaining steps to see the fraction-free inequality solved.
Open the graph, interval notation, and verify tabs to confirm the final answer.
A fraction inequality is a linear inequality that contains one or more fraction coefficients or fraction constants. Examples include x/2 + 1 > 3, where the variable term has a fraction coefficient, and (3/4)x - 2 <= 5, where the coefficient is written as a fraction. The key feature is that the denominator of every fraction is a constant number — not a variable expression.
This is what separates fraction inequalities from rational inequalities. In a rational inequality such as (x + 1)/(x - 2) > 0, the denominator contains a variable and creates undefined points that must be tracked separately. In a fraction inequality, the denominator is always a fixed number, so the only extra step is clearing the fractions before solving.
The standard method for clearing fractions is to multiply both sides of the inequality by the LCD of all denominators. This converts the fraction inequality into a standard linear inequality with integer coefficients, which is then solved with the usual steps. The sign-flip rule still applies at the final division step — and also at the LCD multiplication step if the LCD itself is negative, though that situation is uncommon in typical classroom problems.
For broader first-degree review, the linear inequality calculator covers the same endgame once the fractions are gone.
| Comparison point | Fraction inequality | Rational inequality |
|---|---|---|
| Fraction type | Fraction coefficients — denominator is a constant | Fraction expression — denominator contains a variable |
| Example | x/2 + 1 > 3 | (x+1)/(x-2) > 0 |
| Extra step | Multiply by LCD to clear fractions | Find zeros and undefined points for sign chart |
| Undefined points | None — denominator is always nonzero | Yes — denominator zero creates excluded values |
| Method | LCD clearing then linear solving | Sign chart method |
The LCD clearing method converts any fraction inequality into a standard linear inequality in one step. Once the fractions are gone, the remaining algebra is identical to solving any other linear inequality. Keep the final answer aligned with interval notation so the graph and symbols stay consistent.
List all denominators in the inequality. The LCD is the smallest number that all denominators divide into evenly. For x/2 + x/3 < 4, the denominators are 2 and 3, so the LCD is 6.
Example
Multiply every term on both sides — including whole number terms — by the LCD. This clears all fractions at once. For x/2 + 1 > 3 with LCD 2: multiply to get x + 2 > 6.
Example
If the LCD is positive (which it almost always is, since denominators in classroom problems are positive), the inequality symbol stays the same. If for any reason you multiply by a negative number, flip the symbol.
Example
After clearing fractions, solve the resulting linear inequality with the standard steps: combine like terms, collect variable terms on one side, and divide by the coefficient. Apply the sign-flip rule if the final division is by a negative number.
Example
When you multiply through by the LCD, every term changes — including standalone constants like 1, 2, or 5. Missing those terms breaks the entire inequality.
Use one LCD for the whole inequality. Multiplying one side or one fraction at a time makes it much easier to lose equivalence.
A positive LCD does not flip the symbol, but the final division might. If clearing the fractions leaves a negative coefficient of x, the last division still reverses the inequality.
These examples cover the main classroom patterns: one fraction term, a fraction coefficient, fractions on both sides, and a final sign flip after the LCD step.
The cleanest fraction pattern is one fractional x-term and whole numbers everywhere else. Clear the denominator first, then solve as a standard two-step inequality.
Multiply every term by 2
Before
After
Subtract 2 from both sides
Before
After
Answer
Interval
Set notation
Number line
A fraction coefficient uses the same LCD logic. Multiply every term by 4, then finish the linear isolation with the inclusive symbol preserved.
Multiply every term by 4
Before
After
Add 8 to both sides
Before
After
Divide by 3
Before
After
Answer
Interval
Set notation
Number line
When fractions appear on both sides, find one LCD for the entire inequality. Clear everything at once, then collect the variable terms.
Multiply every term by 6
Before
After
Subtract 2x from both sides
Before
After
Add 6 to both sides
Before
After
Answer
Interval
Set notation
Number line
Clearing the fraction can reveal a negative coefficient of x. The LCD step does not flip the symbol here because 4 is positive, but the final division by -1 does.
Multiply every term by 4
Before
After
Subtract 12 from both sides
Before
After
Divide by -1 and flip the sign
Before
After
Answer
Interval
Set notation
Number line
Once the fractions are cleared and x is isolated, the graph behaves exactly like any other linear inequality. The algebra is front-loaded, but the final visual answer is still a ray or bounded interval with the same open-versus-closed endpoint rule.
For x > 4, draw an open endpoint at 4 and shade to the right. For x <= 8, draw a closed endpoint at 8 and shade to the left. The fraction work changes how you get the boundary value, not how you graph the finished solution.
Strict inequality
Use an open circle when the boundary is excluded, as in x > 4.
Inclusive inequality
Use a closed circle when the boundary is included, as in x <= 8.
Interval notation for a fraction inequality follows the same endpoint rules as every other linear answer. Parentheses exclude the boundary, brackets include it, and infinity always uses a parenthesis.
The only extra care is making sure the boundary value came from the fully cleared inequality. Once the LCD step and any sign flip are handled correctly, translate the final statement directly into interval notation.
The full interval notation guide explains the bracket and parenthesis rules in more detail.
| Inequality | Interval notation | Number line | Meaning |
|---|---|---|---|
| Values greater than a, but not including a. | |||
| Values greater than a, including a. | |||
| Values less than a, but not including a. | |||
| Values less than a, including a. |
A fraction inequality has fraction coefficients with constant denominators (x/2 + 1 > 3). A rational inequality has a variable in the denominator ((x+1)/(x-2) > 0). The solving methods are completely different.
The LCD (least common denominator) is the smallest number all denominators divide into evenly. Multiplying every term by the LCD eliminates all fractions at once, leaving a simpler integer-coefficient inequality to solve.
Only if the LCD is negative. Since denominators in standard fraction inequalities are positive numbers, the LCD is always positive and the symbol stays the same.
Find the LCD of all denominators across both sides, then multiply every term — on both sides — by that LCD. All fractions clear in one step.
Enter mixed numbers as improper fractions or decimals: 1.5 or 3/2. The calculator handles fraction notation with the / symbol.
The rational inequality calculator uses the sign chart method for expressions like (x+1)/(x-2) > 0 where the denominator contains a variable. This calculator uses the LCD clearing method for expressions like x/2 + 1 > 3 where the denominator is a fixed number.
Zone 6
After the LCD method feels natural, the most useful next comparisons are standard linear cleanup, longer multi-step chains, and true rational expressions with variable denominators.
Linear Inequality Calculator
Solve one-variable linear inequalities with steps, interval notation, and a clean number-line graph.
Multi Step Inequality Calculator
Solve multi-step inequalities with distribution, combining like terms, variable collection, and sign-flip detection.
Rational Inequality Calculator
Solve rational inequalities with steps, excluded values, sign charts, and interval notation.