An inequality compares two quantities using \(<\), \(>\), \(\le\), or \(\ge\), such as \(x > 5\). An inequality usually has infinitely many solutions. To graph \(x > 5\), draw an open circle at 5 and shade to the right; for \(x \ge 5\), use a closed circle because 5 is a solution.
Key facts:
- A solution of an inequality is any value that makes it true: \(7\) is a solution of \(x > 5\).
- Open circle for \(<\) and \(>\); closed circle for \(\le\) and \(\ge\).
- Shade right for greater values and left for lesser values.
- “At least” means \(\ge\) and “at most” means \(\le\).
What is an inequality?
An inequality is a statement that compares two quantities that are not necessarily equal. Where an equation uses an equals sign, an inequality uses one of four symbols. For example, \(x > 5\) says “x is greater than 5.”
| Symbol | Meaning | Example |
|---|---|---|
| \(<\) | is less than | \(2 < 7\) |
| \(>\) | is greater than | \(9 > 4\) |
| \(\le\) | is less than or equal to | \(x \le 3\) |
| \(\ge\) | is greater than or equal to | \(x \ge 5\) |
What does a solution of an inequality look like?
A solution of an inequality is any value of the variable that makes it true. For \(x > 5\), the values \(6\), \(5.5\), and \(100\) are solutions because \(6 > 5\), \(5.5 > 5\), and \(100 > 5\). The values \(5\) and \(4\) are not solutions, since \(5 > 5\) is false.
Unlike \(x + 2 = 7\), which has exactly one solution, \(x > 5\) has infinitely many solutions, because there are always more numbers between and beyond those you list.
How do you graph an inequality on a number line?
- Draw a number line and locate the boundary number.
- Draw a circle at the boundary: an open circle for \(<\) or \(>\), because the boundary is not a solution, and a closed circle for \(\le\) or \(\ge\), because it is a solution.
- Shade or draw an arrow toward the solutions: to the right for greater values, to the left for lesser values.
The graph below shows \(x \le 3\): a closed circle at \(3\), shaded to the left.
How do you write an inequality for a real-world situation?
“More than” means \(>\). “Less than” means \(<\). "At least" means \(\ge\). "At most" means \(\le\).
Example: a roller coaster at the county fair requires riders to be at least 48 inches tall. Let \(h\) be a rider’s height in inches. “At least 48” includes 48 itself, so the inequality is \(h \ge 48\). Its graph has a closed circle at \(48\) and shading to the right. A rider who is \(47.5\) inches tall is not a solution, so that rider cannot ride.
Always ask whether the situation limits the values. A number of people must be a whole number, so \(12.5\) people is not a sensible solution even if it satisfies the inequality.
Writing and Graphing Inequalities: frequently asked questions
What is the difference between an equation and an inequality?
An equation uses an equals sign and often has one solution, such as \(x + 2 = 7\). An inequality uses \(<\), \(>\), \(\le\), or \(\ge\) and usually has infinitely many solutions, such as \(x > 5\).
When do you use an open circle and when a closed circle?
Use an open circle when the inequality is \(<\) or \(>\), because the boundary number is not a solution. Use a closed circle when it is \(\le\) or \(\ge\), because the boundary number is a solution.
What does at least mean in an inequality?
At least means the same as or greater than. A sign that says riders must be at least 48 inches tall gives \(h \ge 48\), which includes a rider exactly 48 inches tall.
How many solutions does an inequality have?
Most inequalities with a variable have infinitely many solutions. For \(x < 10\), the numbers \(9\), \(9.9\), \(9.99\), and every smaller number all work, so you cannot list every solution.
How do you check whether a number is a solution?
Substitute the number for the variable and decide whether the statement is true. For \(x \le 3\) and \(x = 3\), \(3 \le 3\) is true, so \(3\) is a solution.
See every chapter on the Grade 6 math page.