How to Solve Inequalities
The best method depends on the structure of the inequality. The key is knowing why each operation changes—or does not change—the statement.
One-Step Inequalities
To solve a one-step inequality, isolate the variable with one inverse operation. For example:
\[ x+6 > 14 \]
Subtract \(6\) from both sides:
\[ x > 8 \]
Adding or subtracting the same value from both sides does not change the inequality direction. The solution includes every value greater than \(8\), not just one answer.
Two-Step Inequalities
A two-step inequality usually requires two inverse operations. Consider:
\[ 4x-9 \leq 15 \]
Add \(9\), then divide by \(4\):
\[ 4x \leq 24 \]
\[ x \leq 6 \]
Since \(4\) is positive, the symbol stays pointed in the same direction. The answer is a set of values satisfying the final inequality.
Linear Inequalities
A linear inequality has a variable to the first power, such as:
\[ 5-3x < 17 \]
Isolate \(x\) using inverse operations, but watch for negative factors. If you multiply or divide both sides by a negative value, reverse the inequality sign.
For instance:
\[ -3x < 12 \]
Dividing by \(-3\) gives:
\[ x > -4 \]
The direction reverses because multiplying \(2 < 5\) by \(-1\) produces \(-2 > -5\). The sign does not reverse after addition, subtraction, multiplication by a positive value, or division by a positive value.
If you are solving an equation rather than comparing ranges of values, try the Solve for x Calculator.
Compound Inequalities
A compound inequality uses either an and condition or an or condition.
An “and” condition requires both statements to be true:
\[ 2 < x \leq 7 \]
Its solution is one continuous interval:
\[ (2,7] \]
An “or” condition allows either statement to be true:
\[ x < -3 \quad \text{or} \quad x > 5 \]
That creates two separate solution regions. Do not combine them into one interval unless every value between the endpoints also works.
Absolute Value Inequalities
For the usual absolute-value rules, assume \(a>0\). Think of \(|x|\) as the distance from zero.
\[ |x| < 4 \]
means \(x\) is less than \(4\) units from zero:
\[ -4 < x < 4 \]
But:
\[ |x| > 4 \]
means \(x\) is more than \(4\) units from zero:
\[ x < -4 \quad \text{or} \quad x > 4 \]
Inclusive symbols include the boundary values. If the comparison value is negative, check the statement before applying these patterns: for example, \(|x| < -2\) has no solution because absolute value cannot be negative.
Rational Inequalities
A rational inequality needs careful sign analysis. Consider:
\[ \frac{x-3}{x+2} > 0 \]
The numerator is zero at \(x=3\). The denominator is zero at \(x=-2\), which is a domain restriction and must always be excluded.
These two values divide the number line into intervals. Test the sign of the entire rational expression on each interval to determine where it is positive or negative. Finding the numerator zero and denominator restriction identifies the boundaries, but the interval sign analysis determines the solution.
If you need help simplifying the numerator or denominator before sign analysis, use the factoring calculator.
Quadratic and Polynomial Inequalities
Quadratic, polynomial, and other nonlinear inequalities often require critical values and interval testing. Solving the related equation finds useful boundaries, but it does not automatically solve the inequality.
For example:
\[ x^2-5x+6 > 0\]
can be written as:
\[ (x-2)(x-3) > 0\]
The roots \(2\) and \(3\) divide the number line into three intervals. Determine the sign of the original expression on each interval to find where it is positive. If you first need to find the roots of the related quadratic equation, use the Quadratic Equation Calculator.
Graphing Inequalities
For a one-variable inequality, start by marking the endpoint on a number line. Use an open circle for strict inequalities such as \(x>4\), because \(4\) is not included. Use a closed circle for inclusive inequalities such as \(x\geq4\), because \(4\) is included.
Shade toward the values that satisfy the statement. An “or” inequality can have multiple shaded regions, while a two-sided “and” inequality may shade only the space between two endpoints.
Writing the Solution Set
You can write a solution set in inequality notation or interval notation. These two statements mean the same thing:
\[ x > 4 \]
\[ (4,\infty) \]
The first uses an inequality symbol; the second uses interval notation. A number line can show the same set with an open circle at \(4\) and shading to the right.
Unlike an equation solved by one value, an inequality may have one interval, several intervals, all real numbers, no solutions, or another appropriate set of values.
Concept Explainer
An inequality compares two expressions using:
\[ <,\quad >,\quad \leq,\quad \geq \]
An equation asks when two expressions are equal. An inequality asks which values make one expression greater than, less than, or equal to another in the stated direction.
For example:
\[ x > 3 \]
is true for every value greater than \(3\). In contrast:
\[ x \geq 3 \]
also includes \(3\). That difference determines whether the endpoint is open or closed on a number line.
The negative-number rule matters because order reverses when you multiply or divide by a negative value. For example:
\[ -2x \leq 8 \]
becomes:
\[ x \geq -4 \]
after dividing both sides by \(-2\).