An inequality is a mathematical statement that compares two values using symbols like <, >, ≤, or ≥ instead of an equals sign. Writing one comes down to choosing the correct symbol, placing it between the right expressions, and understanding what the result means. Whether you’re translating a word problem into math or graphing a solution on a number line, the process follows a few straightforward rules.
The Four Inequality Symbols
Every inequality uses one of four symbols. Two are called “strict” inequalities because they exclude the boundary value, and two are “non-strict” because they include it.
- < (less than) means the value on the left is smaller than the value on the right. Example: 3 < 7.
- > (greater than) means the value on the left is larger than the value on the right. Example: 10 > 4.
- ≤ (less than or equal to) means the value on the left is smaller than or exactly equal to the value on the right. Example: x ≤ 5 means x could be 5, 4, 3, or anything below 5.
- ≥ (greater than or equal to) means the value on the left is larger than or exactly equal to the value on the right. Example: x ≥ 12 means x could be 12, 13, 14, or anything above 12.
A quick way to remember direction: the small, pointed end of the symbol always faces the smaller value, and the wide, open end faces the larger value.
Translating Words Into Inequalities
Most inequality problems start as English sentences, and the trickiest part is figuring out which symbol a phrase maps to. Here are the most common translations:
- “more than” or “greater than” → use >
- “fewer than” or “less than” → use <
- “at least” or “no fewer than” → use ≥
- “at most” or “no more than” → use ≤
The phrases “at least” and “at most” trip up a lot of students because they include the boundary number. “You must be at least 16 to drive” means 16 is allowed, so you’d write age ≥ 16. “The bag holds at most 50 pounds” means 50 is allowed, so you’d write weight ≤ 50.
A Step-by-Step Example
Suppose the problem says: “A student needs more than 80 points to earn a B.” Start by identifying the variable. Let p represent the student’s points. “More than” tells you to use the strict greater-than symbol. The inequality is p > 80. Notice that 80 itself is not included, because the problem says “more than,” not “at least.”
Now change the problem slightly: “A student needs at least 80 points to earn a B.” The phrase “at least” includes 80, so the inequality becomes p ≥ 80.
Writing Compound Inequalities
Sometimes a value falls between two boundaries. If a thermostat keeps a room between 65 and 75 degrees (inclusive), you can write that as a compound inequality: 65 ≤ T ≤ 75, where T is the temperature. This means T is greater than or equal to 65 and less than or equal to 75 at the same time.
If one or both boundaries are not included, swap the ≤ for <. A problem that says “the temperature stays above 65 but below 75” would be written 65 < T < 75, excluding both endpoints.
Graphing Inequalities on a Number Line
A number line graph gives you a visual way to show every value that satisfies an inequality. The two things you need to decide are the type of circle at the boundary and the direction of the shading.
For strict inequalities (< or >), draw an open circle at the boundary number. The open circle signals that the boundary itself is not part of the solution. For non-strict inequalities (≤ or ≥), draw a filled-in (closed) circle to show the boundary is included.
After placing the circle, shade in the direction of the solutions. If x > 5, shade to the right of 5 because the solutions are all numbers larger than 5. If x < 5, shade to the left. For a compound inequality like 2 ≤ x ≤ 8, place closed circles on both 2 and 8 and shade the segment between them.
Writing Solutions in Interval Notation
Interval notation is a shorthand you’ll see in algebra and beyond. Instead of writing x > 3, you write (3, ∞). Instead of x ≤ 10, you write (−∞, 10]. The rules mirror the open-circle and closed-circle logic from graphing.
- Parentheses ( ) mean the boundary is not included, just like an open circle. Use a parenthesis for strict inequalities (< or >).
- Square brackets [ ] mean the boundary is included, just like a closed circle. Use a bracket for non-strict inequalities (≤ or ≥).
- Infinity (∞ or −∞) always gets a parenthesis, never a bracket, because infinity is not an actual number you can reach or include.
Here’s how common inequalities look in all three formats:
- x > 3 → open circle at 3, shade right → (3, ∞)
- x ≤ 10 → closed circle at 10, shade left → (−∞, 10]
- 2 ≤ x < 7 → closed circle at 2, open circle at 7, shade between → [2, 7)
Solving a Simple Inequality
Solving an inequality works almost exactly like solving an equation. You isolate the variable using addition, subtraction, multiplication, or division. There is one critical difference: when you multiply or divide both sides by a negative number, you must flip the inequality symbol.
For example, solve −2x > 8. Divide both sides by −2 to isolate x. Because you’re dividing by a negative, flip the > to <. The result is x < −4. You can check by plugging in a number less than −4, like −5: −2(−5) = 10, and 10 is indeed greater than 8.
If you forget to flip the symbol, your solution set will be the exact opposite of the correct answer. This is the single most common mistake when working with inequalities, so it’s worth making a habit of checking whether you multiplied or divided by a negative at any step.
Putting It All Together
Try a full example from start to finish. A gym charges a $25 membership fee plus $10 per class. You want to spend no more than $75 total. Write and solve the inequality.
Let c represent the number of classes. Total cost is 25 + 10c, and “no more than $75” translates to ≤ 75. The inequality is 25 + 10c ≤ 75. Subtract 25 from both sides: 10c ≤ 50. Divide both sides by 10: c ≤ 5. You can take at most 5 classes. On a number line, place a closed circle at 5 and shade to the left (through 0, since negative classes don’t make sense, your practical answer is 0 through 5). In interval notation for the pure math solution, that’s (−∞, 5], though in context you’d limit it to whole numbers from 0 to 5.