Learning how to solve inequalities means more than isolating a variable: you also need to represent every value that makes the statement true. This guide explains a reliable step-by-step method for one-step, multi-step, compound, and absolute-value inequalities, then shows how to graph solutions on a number line, write them in interval notation, and check your work.
Overview
An inequality compares two expressions using a symbol such as less than (<), greater than (>), less than or equal to (≤), or greater than or equal to (≥). Unlike a typical equation, an inequality usually has a range of possible answers rather than one value.
For example, the solution to x + 3 > 7 is x > 4. Every number greater than 4 works, including 4.1, 10, and 1,000. The number 4 itself does not work because the original statement is strictly greater than, not greater than or equal to.
A dependable workflow for solving inequalities is:
- Simplify both sides when necessary.
- Use inverse operations to isolate the variable.
- Reverse the inequality symbol if you multiply or divide by a negative number.
- Write the solution clearly in inequality form.
- Graph the solution and, when requested, convert it to interval notation.
- Check a value from the proposed solution set in the original inequality.
This process is similar to solving a linear equation, but the sign-reversal rule and the idea of a solution interval require special attention. For a broader review of algebraic isolation, see this step-by-step guide to linear equations.
What to track
One-step inequalities
Use the inverse operation on both sides, just as you would with an equation.
Example 1: x - 5 ≤ 2
Add 5 to both sides:
x ≤ 7
The solution includes 7 and every number less than 7.
Example 2: -3x > 12
Divide both sides by -3. Because you divide by a negative number, reverse the inequality:
x < -4
Forgetting this reversal is one of the most common inequality errors. A quick check confirms the result: choose x = -5. Then -3(-5) = 15, and 15 > 12 is true.
Multi-step inequalities
Follow the usual order of operations. Combine like terms, distribute when needed, and move variable terms to one side before isolating the variable.
Example: 4x - 6 > 2x + 8
- Subtract 2x from both sides: 2x - 6 > 8.
- Add 6 to both sides: 2x > 14.
- Divide by 2: x > 7.
If simplifying produces a statement with no variable, interpret that statement carefully. For example, 3 < 8 is always true, so every real number is a solution. A statement such as 9 ≤ 4 is false, so there is no solution.
Compound inequalities
A compound inequality combines two conditions. In a “between” statement, such as 2 < x ≤ 6, x must satisfy both conditions at the same time. The solution is every number greater than 2 and less than or equal to 6.
To solve a compound inequality written as a chain, perform the same operation on all three parts:
1 < 2x + 3 ≤ nine
Subtract 3 throughout:
-2 < 2x ≤ 6
Divide throughout by 2:
-1 < x ≤ 3
Some compound inequalities use or instead of and. For example, x < -2 or x ≥ 4 describes two separate parts of the number line. Do not combine these into one continuous interval; graph each branch separately.
Absolute-value inequalities
Absolute value measures distance from zero, so absolute-value inequalities often become compound inequalities.
For a positive number a:
- |x| < a means -a < x < a.
- |x| ≤ a means -a ≤ x ≤ a.
- |x| > a means x < -a or x > a.
- |x| ≥ a means x ≤ -a or x ≥ a.
For example, |x - 2| ≤ 5 becomes -5 ≤ x - 2 ≤ 5. Adding 2 throughout gives -3 ≤ x ≤ 7.
Graphing and interval notation
Graphing makes the solution set visible. Use an open circle for a strict inequality (< or >), because the endpoint is not included. Use a closed circle for an inclusive inequality (≤ or ≥), because the endpoint is included. Shade left for values less than the endpoint and right for values greater than the endpoint.
| Algebraic form | Number-line graph | Interval notation |
|---|---|---|
| x < 4 | Open circle at 4; shade left | (-∞, 4) |
| x ≤ 4 | Closed circle at 4; shade left | (-∞, 4] |
| x > 4 | Open circle at 4; shade right | (4, ∞) |
| x ≥ 4 | Closed circle at 4; shade right | [4, ∞) |
| -2 < x ≤ 5 | Open at -2, closed at 5; shade between | (-2, 5] |
Parentheses indicate an excluded endpoint, while brackets indicate an included endpoint. Infinity always uses a parenthesis because infinity is not a real number that can be included in a solution set.
Cadence and checkpoints
When practicing inequalities, pause at consistent checkpoints rather than waiting until the final answer to look for mistakes.
- After simplifying: Check that distribution and combining like terms were done correctly.
- Before dividing: Identify whether the divisor is negative. If it is, plan to reverse the inequality symbol.
- After isolating the variable: Compare the direction of the symbol with the operation used.
- After graphing: Confirm that the circle style matches inclusion and that shading points in the correct direction.
- Before submitting: Test one value inside the shaded region and, when useful, one value outside it.
For a recurring homework set, track the type of error rather than only the number of incorrect answers. Mark mistakes as sign reversal, endpoint choice, shading direction, distribution, or checking error. This creates a focused review list for the next practice session.
How to interpret changes
The form of the answer tells you how the solution behaves. A single inequality such as x > 6 creates a ray extending indefinitely to the right. A bounded compound inequality such as -1 ≤ x < 3 creates a finite interval. An “or” compound inequality creates two disconnected rays or intervals.
Pay particular attention when a solution changes after multiplying or dividing by a negative quantity. For example, multiplying x < 2 by -4 gives -4x > -8, not -4x < -8. The reversal preserves the set of values that make the statement true.
Checking a test value is especially useful for graphs. If your graph says x < 3, try x = 0. The original inequality should be true. Then try a value such as x = 5; it should be false. This check can reveal a reversed shading direction even when the algebra looks correct.
If an online inequality solver gives an unexpected result, enter the original expression carefully and verify parentheses, negative signs, and the inequality symbol. A tool can help confirm a solution, but writing each transformation yourself makes it easier to locate the exact step that changed the answer. For related troubleshooting, see the equation solver troubleshooting guide.
When to revisit
Revisit this guide whenever a new assignment introduces a different inequality format, especially compound or absolute-value inequalities. It is also worth returning before a quiz or exam to practice the sign-reversal rule, open and closed endpoints, and interval notation together.
Use this short final checklist:
- Did you apply the same operation to both sides, or to all parts of a chained inequality?
- Did you reverse the symbol after multiplying or dividing by a negative number?
- Does the endpoint belong in the solution? Choose an open or closed circle accordingly.
- Does the shading direction match less than or greater than?
- Does interval notation use parentheses and brackets correctly?
- Does a test value satisfy the original inequality?
For continued algebra review, connect inequality practice with resources on solving equations step by step and choosing an appropriate equation-solving method. With the checkpoints above, you can use an inequality solver as a verification tool while still understanding why each answer is correct.
