Distributing and Combining Like Terms in Inequalities
Distributing and combining like terms in inequalities is a Grade 7 algebra skill in Big Ideas Math, Course 2. When an inequality involves parentheses and multiple variable terms, first apply the distributive property, then collect all variable terms on one side and constants on the other. For example, 2(x + 3) + x < 15 expands to 2x + 6 + x < 15, which combines to 3x + 6 < 15, and solves to x < 3. Like terms—terms with the same variable raised to the same power—must be combined before isolating the variable. This multi-step process mirrors equation solving except for the inequality sign reversal rule with negatives.
Key Concepts
Property Before you can begin moving terms across the inequality symbol, you must simplify each side independently. This is the "cleanup" phase:.
Step 1: Use the Distributive Property to remove any parentheses. Step 2: After distributing, combine any like terms on the same side of the inequality to finish simplifying the expression.
Examples Example 1 (Distributing): Simplify the inequality $8 2(x + 3) \leq 14$. First, distribute the 2 to get $8 2x 6 \leq 14$. Then, combine the constant like terms (8 and 6) to get $ 2x + 2 \leq 14$. Now it is ready to solve! Example 2 (Multiple Distributions): Simplify $4(x 8) (x + 3) 10$. Distribute the 4 and the negative sign: $4x 32 x 3 10$. Combine like terms ($4x$ with $ x$, and 32 with 3) to get $3x 35 10$. Example 3: Simplify $4(3n + 7) + 5(n 2) < 50$. Distribute to get $12n + 28 + 5n 10 < 50$. Combine like terms to get $17n + 18 < 50$.
Common Questions
How do you solve an inequality like 2(x + 3) + x < 15?
Distribute: 2x + 6 + x < 15. Combine like terms: 3x + 6 < 15. Subtract 6: 3x < 9. Divide by 3: x < 3.
What are like terms in an inequality?
Like terms have the same variable raised to the same power. For example, 2x and 5x are like terms; 2x and 2x² are not.
Why must you combine like terms before isolating the variable?
Combining like terms simplifies the inequality to a form where inverse operations can isolate the variable in the fewest steps.
Does the inequality sign change when combining like terms or distributing?
No—the inequality sign only reverses when you multiply or divide both sides by a negative number. Distribution and combining like terms do not change the direction.
How do you handle variables on both sides of the inequality?
Use addition or subtraction to move all variable terms to one side, then move constants to the other side before dividing to solve.
What step comes first: distribution or combining like terms?
Distribution comes first to remove parentheses. Then combine like terms on the same side before applying inverse operations to isolate the variable.