An inequality compares values using symbols such as <, >, ≤ or ≥. Instead of saying two expressions are equal, it says one is greater, less, or at least/at most another, as in x > 5. An inequality with a variable usually has infinitely many solutions, shown as a ray on a number line.
In Spanish
desigualdad
Also called
comparison statement, inequalities, linear inequality
An open dot leaves the endpoint out; a closed dot includes it. The arrow shows that every number in that direction is also a solution, so there are infinitely many.
Formal definition and notation
An inequality is a statement that two expressions are related by one of the order relations<, >, ≤, ≥ (or ≠). A solution is a value of the variable that makes the statement true; the set of all solutions is the solution set. You solve a linear inequality with the same moves as an equation, except that multiplying or dividing both sides by a negative number reverses the direction of the inequality sign.
The symbol’s open side faces the larger value: in 3 < 8, the wide side opens toward 8.
On a number line, an open dot means the endpoint is not included (<, >); a closed dot means it is included (≤, ≥).
Word clues: “at least” means ≥, “at most” or “no more than” means ≤, “more than” means >, “fewer than” means <.
Check a solution by testing a value from the solution set in the original inequality: for x > 5, try x = 6.
x > 5 and 5 < x say the same thing; read from the variable’s side to avoid flipping by accident.
Rules for solving an inequality
Rules for solving an inequality
If a < b and c < 0, then ac > bc
add or subtract c
sign stays the same: x − 4 < 6 gives x < 10
multiply or divide by c > 0
sign stays the same: 3x ≥ 12 gives x ≥ 4
multiply or divide by c < 0
flip the sign: −2x < 6 gives x > −3
Why the flip: 2 < 5, but multiplying both sides by −1 gives −2 and −5, and −2 > −5. Negating reverses the order on the number line.
Try it: graph an inequality
Interactive
The graph starts as x ≥ −2: a filled dot and an arrow to the right. Drag the endpoint, switch it open or closed, or turn either infinite end on or off to graph x < 3 or −2 ≤ x < 3.
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
Inequality examples, step by step
Example 1: A one-step inequality
Warm-up
Solve x + 7 ≥ 12 and describe the graph.
Subtract 7 from both sides: x ≥ 5.
Check a value: x = 6 gives 13 ≥ 12, true. Check the endpoint: x = 5 gives 12 ≥ 12, true, so 5 is included.
Graph: a closed dot at 5 and a ray to the right.
Answer:x ≥ 5: a closed dot at 5 with an arrow to the right.
Example 2: Dividing by a negative
Core
Solve −3x + 4 < 19.
Subtract 4 from both sides: −3x < 15.
Divide both sides by −3. Dividing by a negative reverses the sign: x > −5.
Check: x = 0 gives 4 < 19, true. x = −6 gives 22 < 19, false, as expected.
Answer:x > −5
Example 3: Stay within a budget
Test-style
A phone plan costs 25 dollars a month plus 10 cents per text. Maya can spend at most 40 dollars a month. What is the greatest number of texts she can send?
Let t be the number of texts. “At most 40” means ≤ 40: 25 + 0.10t ≤ 40.
An equation says two expressions are equal and usually has one solution: x + 2 = 5 means x = 3. The inequality x + 2 > 5 has infinitely many: every number greater than 3.
A compound inequality joins two conditions, such as −2 ≤ x < 3 (a segment between two endpoints). A simple inequality like x > 2 has one endpoint and a ray.
An expression like 2x + 1 has no comparison symbol, so it is not true or false. Adding > 7 turns it into an inequality you can solve.
Quick check: 3 questions
Question 1 of 3
Solve 2x − 3 > 7.
Show answer and explanation
Answer: B. Add 3: 2x > 10. Divide by 2 (positive, so no flip): x > 5. Choice A subtracts 3 instead of adding, choice C flips the sign without a reason, and choice D forgets to divide by 2.
Question 2 of 3
A van holds at most 12 people. Which inequality describes the number of people p in the van?
Show answer and explanation
Answer: D. “At most 12” allows 12 and anything less, so p ≤ 12. p < 12 wrongly leaves out a full van, and the two ≥ and > choices describe at least or more than 12.
Question 3 of 3
Solve −4x ≥ 20.
Show answer and explanation
Answer: A. Divide by −4 and flip the sign: x ≤ −5. Choice B forgets to flip, and choices C and D lose the negative sign. Check: x = −6 gives 24 ≥ 20, true.
Inequality: frequently asked questions
Why do you flip the inequality sign when you multiply or divide by a negative?
Multiplying by a negative reflects numbers across 0 on the number line, which reverses their order. 3 < 7 is true, but after multiplying both sides by −1 you get −3 and −7, and −3 is now the larger one: −3 > −7. Flipping the sign keeps the statement true. Adding or subtracting never flips it.
What is the difference between an open circle and a closed circle?
An open circle on a number line means the endpoint is not part of the solution; use it for < and >. A closed (filled) circle means the endpoint is included; use it for ≤ and ≥. For x < 4 the circle at 4 is open, because 4 is not less than 4. For x ≤ 4 it is closed.
How many solutions does an inequality have?
Usually infinitely many. x > 2 is true for 3, 2.5, 2.01, 100 and every other number greater than 2, which is why the graph is a ray. Some inequalities have no solution, such as x + 1 < x, and some are true for every number, such as x + 1 > x. In word problems, only whole numbers may make sense.
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