Enter one inequality in x. The calculator finds its boundary values, tests the intervals they create, and shows the solution in interval notation and on a number line.
Result
All real values less than 4 satisfy the inequality
x<4⟺(−∞,4)
Conditions
The variable x is real and the comparison is strict.
The boundary x = 4 is excluded because it makes both sides equal.
Only one linear or quadratic comparison is entered at a time.
Steps
Remove the constant Subtract 3 from both sides without changing the direction.2x<8
Isolate x Divide by positive 2, so the less-than sign stays the same.x<4
Translate the boundary Four is not included; the interval extends without bound to the left.(−∞,4)
✓
Independent check
At x = 4, the original comparison is 11 < 11, which is false. At x = 0 it is 3 < 11, true; at x = 5 it is 13 < 11, false.
Scope
What this inequality covers
For a linear inequality, isolate x and reverse the comparison only when multiplying or dividing by a negative number. For a quadratic inequality, first find the zeros, then determine which intervals make the original comparison true.
Isolate a variable in a linear inequality
Keep track of every operation, including the sign reversal required by a negative multiplier or divisor.
Examples: 2x + 3 < 11, -3x + 6 ≥ 15
Find the sign intervals of a quadratic
Use its real zeros as boundaries, then test the expression between and beyond them instead of assuming that both roots are the answer.
Examples: x² - 4 ≤ 0, x² - 5x + 6 > 0
Distinguish included from excluded endpoints
Use closed endpoints for ≤ or ≥ when equality is permitted; use open endpoints for < or >.
Examples: x ≤ -3, x < 5
Recognize empty or all-real answers
When no boundary interval passes the test, report no real solution; when every value passes, report all real numbers.
Examples: x² + 1 < 0, x² + 1 > 0
How to use it
Enter enough information for one clear task
1
Enter a single comparison
Use x and one of <, ≤, >, or ≥. Parentheses make distribution clear. For two joined comparisons, use the compound inequality calculator instead.
2
Inspect the boundary values
For a line, look for the point where the sides are equal. For a quadratic, locate its real zeros before deciding which intervals work.
3
Check one value in each region
Substitute into the original inequality, not only a transformed line. Check a boundary separately when the sign includes equality.
Worked inputs
Examples to try
Use these examples to recognize the method, compare equivalent forms, and check your own work.
Positive linear coefficient
Subtract 3 and divide by positive 2; equality at 4 is excluded.
2x+3<11
Expected result
x<4⟺(−∞,4)
Negative coefficient reverses the sign
Subtract 6, then divide by negative 3 and reverse the comparison.
−3x+6≥15
Expected result
x≤−3⟺(−∞,−3]
Variable on both sides
Distribute, collect x-terms, and keep the inclusive boundary.
2(x−1)≤x+3
Expected result
x≤5⟺(−∞,5]
Quadratic true between its zeros
The zeros are -2 and 2; the upward-opening quadratic is nonpositive between them.
x2−4≤0
Expected result
−2≤x≤2⟺[−2,2]
Quadratic true outside its zeros
Factor as (x - 2)(x - 3); the product is positive outside the two roots.
x2−5x+6>0
Expected result
x<2orx>3⟺(−∞,2)∪(3,∞)
No real solution
A real square is never negative, so x² + 1 is always positive.
x2+1<0
Expected result
∅
Every real value works
Cancel 4x from both sides; the remaining statement 1 ≤ 3 is always true.
4x+1≤4x+3
Expected result
x∈R⟺(−∞,∞)
Complete example
Use a sign chart for a quadratic inequality
A quadratic has two boundary points here. Finding the roots is only the first step; the sign on each interval decides the answer.
2x2+x−6≤0
1
Find the zeros
Factor the polynomial and solve each factor equal to zero.
2x2+x−6=(2x−3)(x+2),x=−2,23
2
Test the three regions
The product is positive left of -2, negative between -2 and 3/2, and positive right of 3/2.
f(−3)=9>0,f(0)=−6<0,f(2)=4>0
3
Include both equality points
The comparison allows zero, so both roots belong to the solution.
−2≤x≤23
[−2,23]
Verification: Substituting -2 and 3/2 into 2x² + x - 6 gives 0; independent region checks at -3, 0, and 2 give 9, -6, and 4, so only the middle region plus its endpoints passes.
Avoidable errors
Common mistakes and how to fix them
Keeping the sign after dividing by a negative
Problem: From -3x ≥ 9, write x ≥ -3.
Why it matters: Multiplication by a negative reverses the order on the real line.
Better approach: Reverse the comparison to x ≤ -3, then substitute a value from each side.
Treating quadratic roots as the whole answer
Problem: For x² - 4 ≤ 0, report only x = -2 and x = 2.
Why it matters: An inequality asks for every value in the regions where the expression has the required sign.
Better approach: Test the intervals created by the roots and include the roots when equality is allowed.
Closing a strict boundary
Problem: Write x ≤ 4 for 2x + 3 < 11.
Why it matters: At 4 the two sides are equal, but a strict less-than sign rejects equality.
Better approach: Evaluate the original comparison at the boundary and use an open endpoint when it fails.
Combining two conditions as one
Problem: Enter an AND or OR question and use only the first comparison.
Why it matters: An intersection or union changes the final set even when each simple inequality is solved correctly.
Better approach: Use the compound inequality calculator for joined or chained comparisons.
Trust the result for the right reasons
Checks, assumptions, and limits
How results are checked
Boundary values come from equality, but membership is determined by the original comparison.
For quadratics, a value from every interval between real roots is checked before a region is selected.
Strict and inclusive signs determine whether a boundary is open or closed.
The displayed interval and number line represent the same solution set.
When to stop and revise the input
One real-variable polynomial inequality of degree at most 2 is supported.
Variable denominators, radicals, absolute values, and functions such as log or sin are outside this input model.
Use the absolute-value or compound calculator when the mathematical structure calls for it.
A numerical-looking graph is a visualization of the algebraic solution, not a substitute for exact boundaries.
Common questions
Solve a linear or quadratic inequality FAQ
When do I reverse the inequality sign?
Reverse it when multiplying or dividing both sides by a negative number. Adding or subtracting the same amount, or dividing by a positive number, does not reverse it.
Why does a quadratic inequality have two intervals sometimes?
Its real zeros split the number line into regions. For an upward-opening quadratic that is greater than zero, the two outer regions may both work, producing a union of intervals.
What do brackets and parentheses mean in interval notation?
A bracket includes a finite endpoint, while a parenthesis excludes it. Infinity is never an attainable endpoint, so its side always uses a parenthesis.
What if the answer is no solution or all real numbers?
Both are valid outcomes. For example, x² + 1 < 0 has no real solution, while x² + 1 > 0 holds for every real x.
Can I solve two inequalities joined by AND or OR here?
Use the compound inequality calculator for joined or chained conditions. This page focuses on one ordinary comparison and the sign intervals it defines.
Sources and curriculum alignment
This page follows standard introductory mathematics notation and learning sequences. Use these references to continue with a complete course treatment.
