Math · College algebra · Concept
Linear inequalities and interval notation
Solve a linear inequality like an equation, with one extra rule: multiplying or dividing both sides by a negative number reverses the inequality sign. The solution is usually a whole interval of numbers, written as an inequality, on a number line or in interval notation.
What an inequality asks
An inequality such as 2x − 5 < 7 asks for every number that makes it true, not just one. Its solution is usually a whole interval: here, every number less than 6. The strict signs < and > leave the endpoint out; ≤ and ≥ include it.
Steps that keep every solution
Adding or subtracting the same number on both sides keeps exactly the same solutions, and so does multiplying or dividing both sides by the same positive number. These are the moves you already use for equations, and none of them changes the direction of the sign.
Why a negative number reverses the sign
Multiplying by −1 reflects the number line through zero, so the order of two numbers reverses: 2 < 5, but −2 > −5. The same happens when you multiply or divide both sides of an inequality by any negative number, so the sign must turn around.
Can you avoid the flip?
Yes: collect the x terms on the side where their coefficient is positive. In 4 − 3x ≤ 10, add 3x to both sides and subtract 10 to get −6 ≤ 3x, so −2 ≤ x. That is x ≥ −2, the same answer as dividing −3x ≤ 6 by −3 and reversing the sign.
Interval notation
Interval notation names a solution set by its endpoints, smaller first. A parenthesis leaves an endpoint out and a square bracket includes it. Infinity is not a number you can reach, so it always takes a parenthesis. A union sign, ∪, joins two separate pieces.
| Inequality | Interval notation | On a number line |
|---|---|---|
| x < 6 | (−∞, 6) | Open circle at 6, shaded to the left |
| x ≥ −2 | [−2, ∞) | Filled dot at −2, shaded to the right |
| −1 < x ≤ 4 | (−1, 4] | Open at −1, filled at 4, shaded between |
| x ≤ 0 or x > 3 | (−∞, 0] ∪ (3, ∞) | Two rays pointing apart |
Compound inequalities
A compound inequality joins two conditions. “And” keeps the numbers that satisfy both, usually one interval: solve a three-part inequality such as −3 ≤ 2x + 1 < 7 by doing the same thing to all three parts. “Or” keeps the numbers that satisfy either condition, which gives two pieces joined by ∪.
When the inequality is not linear
For a quadratic inequality, move every term to one side and factor. The expression can change sign only at its zeros, so a sign chart of the intervals between them gives the answer. x² − x − 6 = (x − 3)(x + 2) is negative only between −2 and 3.
Check with test values
Substitute one number from inside your answer and one from outside it into the original inequality: the first must make it true and the second false. Test an endpoint too. It belongs to the answer only when the original sign allows equality.
Common mistakes
- Forgetting to reverse the sign after dividing by a negative: −3x ≤ 6 gives x ≥ −2, not x ≤ −2.
- Reversing the sign after adding or subtracting a negative number: only multiplying or dividing by a negative reverses it.
- Putting a bracket at infinity, as in [2, ∞]: infinity always takes a parenthesis.
- Writing the endpoints in the wrong order, as in (6, −∞): the smaller endpoint comes first.
Key terms
- Inequality
- A comparison using <, >, ≤ or ≥. Strict comparisons exclude equality; non-strict comparisons allow it. Multiplying both sides by a negative number reverses the direction.
- Interval
- An unbroken stretch of the number line, such as all x from 2 to 5. A bracket [ ] includes an endpoint and a parenthesis ( ) leaves it out.
- Open interval
- An interval that leaves out its endpoints: (a, b) means a < x < b. Infinity always takes a parenthesis because it is not a number you can reach.
- Closed interval
- An interval that includes both endpoints: [a, b] means a ≤ x ≤ b. A half-open interval, such as [a, b), includes one endpoint and leaves out the other.
- Compound inequality
- Two inequalities joined by “and” or “or”. “And” keeps the values that satisfy both, an intersection such as −2 ≤ x < 3; “or” keeps the values that satisfy either, a union such as x < −3 or x > 3.
- Sign chart
- A number line split at the zeros and undefined points of an expression, showing where it is positive, negative, zero or undefined. The sign can change only at those points.
Work through an example
Solve 7 − 2x > 3x − 8. Write the solution in interval notation and check it.
Solve a linear inequality with a sign flip →Sources and scope
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Try in the workspace
Open the example inputs, change a value and keep a useful result on your board.
Check the sign flip in Steps, assumptions & inequalities Check each step in Math Open worked example on a board Inequality rules in Math ReferenceYour existing work stays on this device. Examples open as editable copies.