An inequality is two expressions connected by a \(<\), \(>\), \(\le\), or \(\ge\) sign. To solve an inequality means to find all values of the variable for which it becomes a true numerical inequality. The answer will not be a single number, but an entire set: an interval on the number line or a union of intervals. This page is a database common to all types of inequalities: how to read the signs, what transformations are allowed, why the sign flips when dividing by a negative number, and how to write the answer correctly. And then — an overview of all types of inequalities with links to detailed explanations and a practice tool to test yourself.
What it means to solve an inequality
Solving an inequality means finding any value of the variable that makes the inequality a true numerical inequality. And to solve an inequality means to find all such values and write them as a set.
Compare it to an equation. For the equation \(2x = 8\), the answer is a single number \(x = 4\). For the inequality \(2x < 8\), the answer is infinitely many numbers: \(3\), \(0\), \(-100\), and \(3.999\) will all work. It's impossible to list them one by one, so the answer is given as an interval: \(x < 4\), i.e., \((-\infty;\ 4)\).
It's easy to check if a specific number is a solution: substitute it for the variable and see if you get a true inequality.
- \(x = 1\) in the inequality \(2x < 8\): \(2 \cdot 1 = 2 < 8\) — true, so \(1\) is a solution.
- \(x = 5\): \(2 \cdot 5 = 10 < 8\) — false, so \(5\) is not a solution.
Two inequalities are called equivalent if they have the same solution sets. The entire solution of an inequality is a chain of transitions to a simpler equivalent inequality until only \(x\) is left on the left side.
Inequality Signs: Strict and Non-Strict
There are four signs, and the answer and its format depend on which one is used. Strict signs \(<\) and \(>\) do not include the boundary point in the answer, while non-strict \(\le\) and \(\ge\) do.
This distinction carries through the entire solution: a strict sign results in an open (empty) point on the number line and a round bracket in the answer, while a non-strict sign results in a closed point and a square bracket.
| Sign | How it is read and what it means for the answer |
|---|---|
| < | less thanstrictopen point, round bracket |
| > | greater thanstrictopen point, round bracket |
| ≤ | less than or equal tonon-strictclosed point, square bracket |
| ≥ | greater than or equal tonon-strictclosed point, square bracket |
Equivalent Transformations: What You Can Do with an Inequality
You can do almost the same things with an inequality as with an equation, but with one important caveat. You can add and subtract freely: if you add or subtract the same thing from both sides, the solution set doesn't change. This is how terms are 'moved' across the sign, changing their plus to a minus.
However, multiplication and division depend on the sign of the multiplier. When multiplying or dividing both sides by a negative number, the inequality sign changes to the opposite: \(<\) becomes \(>\), \(\le\) becomes \(\ge\), and vice versa. The reason is visible with numbers: from the true \(3 < 5\), after multiplying by \(-1\), we get \(-3\) and \(-5\), and \(-3 > -5\) — the order of the numbers has flipped.
The same happens if you swap the left and right sides: from \(7 < y\), you get \(y > 7\).
| What we do to both sides | What happens to the sign |
|---|---|
| Add or subtract the same number or expression | sign remains |
| Multiply or divide by a positive number | sign remains |
| Multiply or divide by a negative number | sign changes |
| Swap the left and right sides | sign changes |
| Multiply or divide by an expression with a variable (x − 2) | not allowed |
How to Write the Answer: Number Line and Intervals
When the variable is isolated, the solution is almost ready — you just need to format it. The most convenient way is to first mark the answer on the number line, and then derive the interval from the line.
There are only three rules for writing, and they don't change depending on the type of inequality:
- round bracket — the point is not included in the answer (strict sign, open point);
- square bracket — the point is included (non-strict sign, closed point);
- infinity always uses a round bracket: \((-\infty;\ 4)\), \([3;\ +\infty)\) — there is no such notation as \([-\infty;\ 4]\); infinity is not a number and cannot be reached.
If the solution consists of two parts of the line, they are joined by the union symbol \(\cup\): for example, \((-\infty;\ -5) \cup (3;\ +\infty)\).
Types of Inequalities and How to Solve Them
Inequalities are distinguished by how the variable enters them — whether it's to the first power, under a square, in the denominator, or under an absolute value. This is what determines the solution method, so the first step in any solution is to identify the type of inequality. The general logic is the same: transform the inequality into the form 'expression compared to zero', and then apply the appropriate technique for that type. A summary of types and methods is in the table below.
Separately, remember how to handle absolute values — the table doesn't cover this. For \(a > 0\), the inequality \(|f(x)| < a\) is equivalent to the double inequality \(-a < f(x) < a\) (forming a system), and \(|f(x)| > a\) is equivalent to the disjunction \(f(x) < -a\) or \(f(x) > a\) (forming a union of intervals). If \(a < 0\), there's nothing to think about: the absolute value is non-negative, so \(|f(x)| < a\) has no solutions, and \(|f(x)| > a\) is true for all valid \(x\).
Irrational (with roots) and exponential inequalities are covered in higher grades: in addition to standard transformations, they involve working with the domain of permissible values.
Detailed explanations of individual types: linear inequalities, rational inequalities and the interval method, and finding roots of a quadratic trinomial is conveniently done using Vieta's formulas.
| Type | Appearance and solution method |
|---|---|
| Linear | ax + b > 0move terms and divide by the coefficient of x |
| Quadratic | ax2 + bx + c > 0find the roots of the trinomial and look at the parabola |
| Fractional-Rational | P(x) / Q(x) ≥ 0interval method, exclude zeros of the denominator |
| With Absolute Value | | f(x) | < aremove absolute value by definition |
| System | two inequalities in a curly bracesolve each, take the intersection |
Systems and Disjunctions of Inequalities
Often, one inequality breaks down into two, and then it's important not to confuse what to do with the resulting sets.
A system (curly brace) means 'both this and that simultaneously'. Solve each inequality separately, mark both answers on the same number line, and take the intersection — the common part. If there is no common part, the system has no solutions.
A disjunction (square bracket) means 'either this or that'. Here, you take the union — everything shaded by at least one of the answers. A disjunction appears, for example, when removing an absolute value with a 'greater than' sign.
A double inequality of the form \(-3 < 2x + 1 \le 7\) is a special case: it's a concise way of writing a system. Transformations are applied to all three parts simultaneously — subtract, divide, and when dividing by a negative number, flip both signs together.
Examples of Solving Inequalities
Example 1. Checking if a number is a solution
Task. Is \(x = -1\) a solution to the inequality \(3x + 5 \ge 1\)?
Step 1. Substitute \(-1\) for \(x\): \(3 \cdot (-1) + 5 = -3 + 5 = 2\).
Step 2. Compare with the right side: \(2 \ge 1\) — this is a true numerical inequality.
Answer. Yes, \(-1\) is one of the solutions. But solving an inequality means finding all such numbers, not just one: here the answer is \(x \ge -\frac{4}{3}\).
Example 2. Linear inequality with sign change
Task. Solve \(7 - 3x < 1\).
Step 1. Move \(7\) to the right (i.e., subtract \(7\) from both sides — the inequality sign does not change): \(-3x < 1 - 7\), so \(-3x < -6\).
Step 2. Divide both sides by \(-3\). Since the divisor is negative, the sign flips: \(x > 2\).
Step 3. Mark on the number line: an open point at \(2\), shade everything to the right.
Answer. \(x > 2\), which is \(x \in (2;\ +\infty)\).
Example 3. Quadratic inequality
Task. Solve \(x^2 - 4x - 5 \le 0\).
Step 1. Find the roots of the trinomial: \(D = 16 + 20 = 36\), \(\sqrt{D} = 6\), so \(x_1 = -1\), \(x_2 = 5\).
Step 2. The coefficient of \(x^2\) is positive, so the parabola opens upwards: between the roots, the trinomial is negative, and outside the roots, it's positive.
Step 3. We need the sign \(\le 0\), meaning negative values and the roots themselves (non-strict sign) — we take the segment between the roots, including the endpoints.
Answer. \(x \in [-1;\ 5]\).
Example 4. Inequality with absolute value
Task. Solve \(|x + 1| > 4\).
Step 1. The absolute value is greater than a positive number — this means the expression inside the absolute value is either greater than \(4\) or less than \(-4\). This forms a disjunction: \(x + 1 > 4\) or \(x + 1 < -4\).
Step 2. Solve each inequality: from the first, \(x > 3\); from the second, \(x < -5\).
Step 3. Since this is a disjunction, we take the union of the intervals, not the intersection.
Answer. \(x \in (-\infty;\ -5) \cup (3;\ +\infty)\).
Example 5. System of inequalities
Task. Solve the system \(\begin{cases} 2x + 1 > -3 \\ x - 4 \le 0 \end{cases}\)
Step 1. First inequality: \(2x > -4\), divide by positive \(2\) — the sign remains the same, \(x > -2\).
Step 2. Second inequality: \(x \le 4\).
Step 3. Mark both answers on the same number line and take the intersection — the region shaded twice. The left boundary is not included (strict sign), the right boundary is included (non-strict).
Answer. \(x \in (-2;\ 4]\).
Common Mistakes in Solving Inequalities
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Dividing both sides of an inequality by a negative number and keeping the same sign.
When multiplying or dividing both sides by a negative number, the sign must change to the opposite. From \(-5x \ge 10\), you get \(x \le -2\), not \(x \ge -2\). You can check this by substitution: \(x = -3\) gives \(15 \ge 10\) — true, so the correct answer is indeed 'less than or equal to'.
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Writing a square bracket for infinity: $[-\infty;\ 7]$.
Infinity is not a number; it cannot be 'reached', so the bracket around \(\infty\) is always round: \((-\infty;\ 7]\). A square bracket is used only for a finite boundary that is included in the answer.
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Confusing bracket types and point types: using a square bracket for a strict sign.
A strict sign (\(<\), \(>\)) means the boundary point is not included: an open point and a round bracket. A non-strict sign (\(\le\), \(\ge\)) means the boundary point is included: a closed point and a square bracket. The check is simple: substitute the boundary point itself into the original inequality and see if it results in a true statement.
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Multiplying both sides by an expression with a variable to 'get rid of the fraction'.
The sign of such an expression is unknown, so it's unclear whether to flip the inequality sign or not — the transformation is not equivalent. Instead, move everything to one side, combine into a single fraction, and use the interval method.
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Writing a single number as the answer: 'x = 3' instead of a set.
The solution to an inequality is a set of values. Write it as an interval: \(x > 3\), which is \((3;\ +\infty)\). A single number can be an element of the answer, but not the answer itself.
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Taking the union for a system and the intersection for a disjunction.
For a system (curly brace, 'and') — take the intersection, the common part of the intervals. For a disjunction (square bracket, 'or') — take the union using the \(\cup\) symbol. Also, the notation '\(-2 < x > 5\)' is meaningless — such an answer should be written as a union of two intervals.
Questions and Answers
What does it mean to solve an inequality?
To solve an inequality means to find all values of the variable for which it becomes a true numerical inequality, and to write them as a set. The answer will be an interval or a union of intervals, not a single number.
What are the types of inequalities?
Based on the form of the expression with the variable, inequalities are classified as linear (\(ax + b > 0\)), quadratic (\(ax^2 + bx + c > 0\)), fractional-rational (variable in the denominator), with absolute value, irrational, and exponential. Systems and disjunctions of inequalities are also distinguished. Based on the sign, they are divided into strict (\(<\), \(>\)) and non-strict (\(\le\), \(\ge\)).
When does the inequality sign flip?
In two cases: when both sides are multiplied or divided by a negative number, and when the left and right sides are swapped. When adding, subtracting, or dividing by a positive number, the sign remains the same.
When is a bracket round, and when is it square?
Round — if the boundary point is not included in the answer (strict sign, open point on the number line). Square — if it is included (non-strict sign, closed point). The bracket is always round near \(+\infty\) and \(-\infty\).
How does solving an inequality differ from solving an equation?
The transformations are almost the same, but the answer to an equation usually consists of individual numbers, while the answer to an inequality consists of entire intervals. Additionally, an inequality has a rule for changing the sign when multiplying or dividing by a negative number, which equations do not have.
What is considered a fractional inequality and where is it covered?
A fractional (fractional-rational) inequality is one where the variable appears in the denominator, e.g., \(\frac{P(x)}{Q(x)} \ge 0\); due to the denominator, it has a domain of permissible values, and it cannot be solved by simple division. A step-by-step algorithm with examples is provided in a separate material rational inequalities and the interval method.
In what grade are inequalities studied?
Linear inequalities and numerical intervals are studied in 8th grade (in the Merzlyak curriculum, it's 9th grade), quadratic and fractional-rational inequalities with the interval method are studied in 8th-9th grades, and irrational and exponential inequalities are studied in higher grades.
