Inequality Solver

Solve linear, polynomial, rational, and absolute-value inequalities with interval notation, sign charts, and a number line.

Results are calculated automatically as you enter data.

Sign chart steps
SolutionEnter an inequality.

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What Is Solving One-Variable Inequalities?

Solving a one-variable inequality means finding every value of \(x\) that makes a comparison true. Instead of asking for a single value where two expressions are equal, an inequality asks where one expression is less than, less than or equal to, greater than, or greater than or equal to another expression.

For example, the inequality

$$ x^2 - 5x + 6 \le 0 $$

does not have just one answer. It is true for a whole interval of values. The solution set is

$$ [2,3] $$

which means every value from \(2\) through \(3\), including both endpoints.

Inequalities are useful because many real problems are about ranges, limits, thresholds, and conditions. A student may need to know when an expression is nonnegative, a teacher may want to check a sign chart, or a graphing task may require identifying where a function is above or below the \(x\)-axis.

Why Inequalities Matter

Inequalities help describe situations where an exact value is less important than a condition being met. In algebra, they are used to find where functions are positive, negative, above zero, below zero, or within a certain range.

They also prepare students for later topics such as function domains, graph interpretation, optimization, calculus sign analysis, and numerical methods. A sign chart, for example, is a bridge between algebra and graphing: it shows how important points divide the number line into regions where the expression behaves consistently.

Key Terms to Know

  • Inequality: A mathematical statement using \(<\), \(\le\), \(>\), or \(\ge\).
  • Solution set: All values of \(x\) that make the inequality true.
  • Interval notation: A compact way to describe a continuous set of numbers, such as \((1,5]\).
  • Set-builder notation: A way to describe a set by a condition, such as \(\{x \mid x \ge 2\}\).
  • Root or zero: A value of \(x\) where a function equals zero.
  • Critical point: A value that can separate the number line into intervals, usually a root or a point where the expression is undefined.
  • Discontinuity: A point where an expression is not defined or does not behave continuously, such as a zero in the denominator of a rational expression.
  • Sign chart: A table or number-line method that tests whether an expression is positive, negative, or zero on different intervals.
  • Search range: The minimum-to-maximum interval within which this solver reports solutions.

How Solving Inequalities with Sign Charts Works

A common way to solve a one-variable inequality is to move everything to one side and compare the result with zero. If the original inequality has a left side \(L(x)\) and a right side \(R(x)\), define

$$ f(x)=L(x)-R(x) $$

Then the inequality can be rewritten as one of these comparisons:

$$ L(x)
$$ L(x)\le R(x) \Longleftrightarrow f(x)\le 0 $$
$$ L(x)>R(x) \Longleftrightarrow f(x)>0 $$
$$ L(x)\ge R(x) \Longleftrightarrow f(x)\ge 0 $$

After that, the main task is to determine where \(f(x)\) is positive, negative, zero, or undefined.

The sign-chart method usually follows these steps:

  1. Rewrite the inequality so one side is \(0\).
  2. Find the critical points, including roots where \(f(x)=0\) and values where \(f(x)\) is undefined.
  3. Use those critical points to divide the number line into intervals.
  4. Choose a test value inside each interval.
  5. Evaluate the sign of \(f(x)\) at the test value.
  6. Include the intervals where the inequality is true.
  7. Decide whether each endpoint is included or excluded.

For a numerical sign-chart approach, the test value is often a midpoint. If an interval runs from \(a\) to \(b\), the midpoint is

$$ c=\frac{a+b}{2} $$

If \(f(c)\) satisfies the inequality, the interval is included in the solution. If it does not, the interval is excluded.

Endpoint rules matter:

  • A root is excluded for strict inequalities such as \(<\) and \(>\).
  • A root may be included for inclusive inequalities such as \(\le\) and \(\ge\).
  • A point where the expression is undefined is excluded, even if nearby values satisfy the inequality.

For supported polynomial, rational, and non-nested absolute-value expressions, this calculator builds a structured algebraic sign chart (up to degree 12 for each polynomial piece). More complex expressions use adaptive numerical analysis and label any unclassified regions as unresolved. In either mode, displayed decimal endpoints are approximations rather than symbolic fractions or radicals.

Examples of Inequalities in Practice

Example 1: Quadratic Inequality

Solve:

$$ x^2-5x+6\le 0 $$

Factor the expression:

$$ x^2-5x+6=(x-2)(x-3) $$

The roots are \(x=2\) and \(x=3\). These roots divide the number line into three intervals:

Interval Test value Sign of \(f(x)\) Included?
\((-\infty,2)\) \(0\) positive No
\((2,3)\) \(2.5\) negative Yes
\((3,\infty)\) \(4\) positive No

Because the inequality is \(\le 0\), the roots are included. The solution is

$$ [2,3] $$

Example 2: Rational Inequality with an Excluded Point

Solve:

$$ \frac{x-1}{x+2}\ge 0 $$

There are two important points:

  • \(x=1\), where the numerator is zero.
  • \(x=-2\), where the denominator is zero, so the expression is undefined.

These points split the number line into intervals:

Interval Sign of the expression Included?
\((-\infty,-2)\) positive Yes
\((-2,1)\) negative No
\((1,\infty)\) positive Yes

The value \(x=-2\) is never included because the expression is undefined there. The value \(x=1\) is included because the expression equals zero and the inequality allows \(\ge 0\).

The solution is

$$ (-\infty,-2)\cup[1,\infty) $$

If a calculator is searching only from \(-10\) to \(10\), the displayed solution would be clipped to that selected range:

$$ [-10,-2)\cup[1,10] $$

Example 3: Absolute Value Inequality

Solve:

$$ |x-3|\le 2 $$

The expression \(|x-3|\) means the distance between \(x\) and \(3\). The inequality says that distance must be at most \(2\).

That means \(x\) can be as low as \(1\) and as high as \(5\):

$$ 1\le x\le 5 $$

In interval notation, the solution is

$$ [1,5] $$

Both endpoints are included because the inequality uses \(\le\).

Example 4: Empty-Set Case

Some inequalities have no solution in a selected range. For example,

$$ x^2+1<0 $$

has no real solution because \(x^2\) is always nonnegative, so \(x^2+1\) is always positive.

The solution is the empty set:

$$ \varnothing $$

For a structured algebraic result, an empty set means no value in the selected search range satisfies the inequality. For numerical fallback, review the unresolved-regions card before treating an empty verified set as complete.

How to Interpret the Result

The main result is a solution set for \(x\) inside the selected search range. If the result is written in interval notation, the symbols tell you which values are included.

Notation Meaning
\((a,b)\) Values between \(a\) and \(b\), excluding both endpoints
\([a,b]\) Values from \(a\) to \(b\), including both endpoints
\((a,b]\) Excludes \(a\) but includes \(b\)
\([a,b)\) Includes \(a\) but excludes \(b\)
\(\cup\) Union; combines separate intervals
\(\varnothing\) Empty set; no solutions were found in the selected range

A set-builder result such as

$$ \{x \mid f(x)\le 0\} $$

means “all values of \(x\) such that \(f(x)\) is less than or equal to zero.”

Critical points help explain why the solution is divided into intervals. They are not automatically solutions. A critical point might be a valid included root, an excluded root, or a value where the expression is undefined.

The sign chart shows the reasoning behind the interval result. If the test value for an interval satisfies the transformed inequality, the whole interval is treated as part of the solution. If the test value does not satisfy it, that interval is excluded.

Decimal endpoints should be read as approximations of the underlying critical values, even when the interval structure is determined algebraically.

Common Mistakes and Misconceptions

A common mistake is treating an inequality like an equation and expecting only one answer. Most inequalities have intervals as solutions, not isolated values.

Another common mistake is forgetting the difference between strict and inclusive inequalities. For

$$ f(x)<0 $$

a root where \(f(x)=0\) is not included. For

$$ f(x)\le 0 $$

that same root may be included if the expression is defined there.

Rational inequalities create another important issue: zeros of the denominator are excluded. Even if values near a discontinuity satisfy the inequality, the undefined point itself cannot be part of the solution.

It is also easy to confuse a full mathematical solution with a solution inside a selected search range. If the search range is \([-10,10]\), values outside that range are not reported.

For numerical results, do not assume every displayed decimal is exact. A value such as \(1.4142136\) may represent an approximation to an irrational or otherwise non-decimal endpoint.

Input format can also cause mistakes. A single-inequality solver expects one comparison at a time, such as

$$ x^2-5x+6\le 0 $$

A chained inequality such as

$$ 1

should be solved or entered as separate conditions if the tool does not support chained comparisons.

When to Use One-Variable Inequality Solving

Use this kind of inequality solving when you need to:

  • Find where a polynomial is positive, negative, nonpositive, or nonnegative.
  • Solve rational inequalities with excluded denominator values.
  • Check an absolute-value inequality.
  • Convert an inequality solution into interval notation.
  • Understand a sign chart for an algebra class.
  • Visualize a solution set on a number line.
  • Estimate a solution numerically when an exact symbolic method is not practical.
  • Check work over a specific range of \(x\)-values.

Limitations and Things to Keep in Mind

This calculator uses a structured algebraic sign chart for supported polynomial, rational, and absolute-value expressions. More complex functions use adaptive numerical analysis and are explicitly labeled uncertified.

Structured algebraic analysis supports polynomial and rational pieces through degree 12 and non-nested absolute-value branches. Expressions outside those limits automatically use the warned numerical method.

The result is limited to the chosen search range. If a solution exists outside the minimum and maximum bounds, it will not appear in the output.

Numerical fallback cannot guarantee that every oscillation, root, or discontinuity was found. Regions that cannot be classified within the evaluation budget are displayed separately as unresolved instead of being silently included or excluded.

Some endpoints may be displayed as decimals instead of exact fractions, radicals, or symbolic expressions. For example, a true endpoint such as \(\sqrt{2}\) may appear approximately as

$$ 1.4142136 $$

Displayed values may also be rounded. Very small values may be shown as \(0\), and very large or very small magnitudes may be shown in scientific notation.

The method assumes a single variable, \(x\), and a single inequality operator. It does not handle multiple variables, logical combinations, equality-only equations, not-equal comparisons, or chained inequalities unless the interface specifically supports them.

For homework, exams, engineering, scientific, financial, or official work, use the numerical result as a guide and double-check important answers with exact algebra, graphing, or a qualified instructor or professional.

How to Use This Calculator

  1. Enter one inequality in the variable \(x\) using \(<\), \(\le\), \(>\), or \(\ge\).
  2. Enter the minimum and maximum \(x\)-values for the search range. The minimum must be less than the maximum.
  3. Use supported math syntax, such as ^ or ** for powers, |x| or abs(x) for absolute value, and functions such as sqrt, sin, cos, tan, log, or exp when needed.
  4. Review the interval-notation solution.
  5. Check the set-builder form to see the equivalent comparison with zero.
  6. Look at the critical points and sign chart to understand why each interval is included or excluded.
  7. Use the number-line graph to visualize the solution inside the selected range.

Frequently Asked Questions

Why does the solver ask for a minimum and maximum value?

The solver works over a selected finite search range. That range tells it where to look for roots, discontinuities, and intervals that satisfy the inequality. Values outside the range are not included in the result.

What is the difference between parentheses and brackets in interval notation?

Parentheses mean an endpoint is excluded. Brackets mean an endpoint is included. For example, \((1,5]\) means values greater than \(1\) and less than or equal to \(5\).

Why is a denominator zero excluded from the solution?

A rational expression is undefined when its denominator is zero. Since the expression does not have a value at that point, the inequality cannot be true there. That point is excluded even if nearby intervals are included.

Can I enter a chained inequality such as \(1<x<3\)?

A chained inequality should not be entered as one expression if the solver accepts only one comparison operator. Instead, solve the two parts separately or rewrite the condition using a supported form. For example, an absolute-value expression such as \(|x-2|<1\) may describe the same interval in some cases.

Are the displayed endpoints exact?

Interval structure may be determined algebraically, but endpoints are displayed as decimal approximations. If you need symbolic fractions or radicals, verify the displayed decimals with an exact symbolic method.

What does an unresolved region mean?

It means an unsupported expression required numerical fallback and the solver could not classify that region confidently within its evaluation or refinement limits. The verified intervals are partial information, not a complete solution set; use exact analysis or another independent method for that region.

Sources and References

Books and Textbooks

  1. Jay Abramson. College Algebra. OpenStax, 2015. Chapter 2, especially “Key Concepts” and Section 2.7, “Linear Inequalities and Absolute Value Inequalities.” https://openstax.org/books/college-algebra/pages/2-key-concepts
  1. Lynn Marecek and Andrea Honeycutt Mathis. Intermediate Algebra 2e. OpenStax, 2020. Sections 7.6, “Solve Rational Inequalities,” and 9.8, “Solve Quadratic Inequalities.” https://openstax.org/books/intermediate-algebra-2e/pages/7-6-solve-rational-inequalities and https://openstax.org/books/intermediate-algebra-2e/pages/9-8-solve-quadratic-inequalities
  1. Michael T. Heath. Scientific Computing: An Introductory Survey. Revised 2nd ed., SIAM, 2018. Chapter 5, “Nonlinear Equations.” https://epubs.siam.org/doi/book/10.1137/1.9781611975581

Online and Official Sources

  1. Michael T. Heath. “CS 450 – Numerical Analysis, Chapter 5: Nonlinear Equations.” University of Illinois Urbana-Champaign lecture slides based on Scientific Computing: An Introductory Survey, 2019. Used for interval bisection, bracketing, and numerical root-finding limitations. https://heath.cs.illinois.edu/scicomp/notes/cs450_chapt05.pdf