Addition and Subtraction Properties of Inequality
The Addition and Subtraction Properties of Inequality state that adding or subtracting the same number from both sides of an inequality preserves the direction of the inequality sign. If x - 3 > 7, adding 3 to both sides gives x > 10. Unlike the multiplication and division properties, these properties never require flipping the sign. Chapter 2 of OpenStax Elementary Algebra 2E introduces these properties immediately after covering the same properties for equations, making it easy to see the parallel structure. These properties are the building blocks for solving all multi-step inequalities.
Key Concepts
Property Subtraction Property of Inequality For any numbers $a$, $b$, and $c$, if $a < b$, then $a c < b c$. if $a b$, then $a c b c$.
Addition Property of Inequality For any numbers $a$, $b$, and $c$, if $a < b$, then $a + c < b + c$. if $a b$, then $a + c b + c$.
Examples To solve $x + 7 \leq 15$, subtract 7 from both sides. This gives $x \leq 8$. The solution is all numbers less than or equal to 8, or $( \infty, 8]$.
Common Questions
What are the Addition and Subtraction Properties of Inequality?
Adding or subtracting the same number from both sides of an inequality does not change the direction of the inequality. If a > b, then a + c > b + c and a - c > b - c.
Does adding a number to both sides of an inequality ever flip the sign?
No. Addition and subtraction never change the direction of an inequality sign. Only multiplication or division by a negative number flips the sign.
How do I use the addition property to solve an inequality?
Add the opposite of any constant being added to the variable side. For x + 5 < 12, subtract 5 from both sides to get x < 7.
How is the Addition Property of Inequality different from the Multiplication Property?
The addition and subtraction properties always preserve the inequality direction. The multiplication property requires flipping the sign when multiplying or dividing by a negative number.
When do students learn the Addition and Subtraction Properties of Inequality?
These properties are introduced in algebra 1, covered in OpenStax Elementary Algebra 2E Chapter 2.
What is the solution set of x - 4 is greater than or equal to 6?
Add 4 to both sides: x is greater than or equal to 10. The solution set is all real numbers 10 or larger.
How do these properties relate to properties of equality?
They are analogous: the same operation applied to both sides of an equation keeps it balanced. For inequalities, addition and subtraction similarly preserve the relationship, while multiplication by a negative requires extra care.