Solving Two-Step Inequalities with Distributive Property
Solving two-step inequalities with the distributive property is a Grade 7 algebra skill in Big Ideas Math, Course 2. When an inequality contains parentheses, first apply the distributive property to expand the expression—for example, 3(x + 2) > 9 becomes 3x + 6 > 9. Then isolate the variable with inverse operations: subtract 6 from both sides to get 3x > 3, then divide by 3 to get x > 1. Critical rule: dividing or multiplying both sides by a negative number reverses the inequality sign. Solutions are written in inequality notation and graphed on a number line with open or closed circles.
Key Concepts
When solving inequalities containing parentheses, first apply the distributive property: $a(b + c) = ab + ac$, then solve using the standard two step process. Remember to reverse the inequality symbol when multiplying or dividing by a negative number.
Common Questions
What is the first step when solving an inequality with parentheses?
Apply the distributive property to expand the expression. Multiply the factor outside the parentheses by each term inside before applying inverse operations.
How do you solve 3(x + 2) > 9?
Distribute: 3x + 6 > 9. Subtract 6: 3x > 3. Divide by 3: x > 1. Graph with an open circle at 1 and an arrow pointing right.
When do you flip the inequality sign?
Only when you multiply or divide both sides by a negative number. For example, −2x < 8 becomes x > −4 after dividing by −2.
How do you graph the solution to x > 1 on a number line?
Draw an open circle at 1 (since 1 is not included) and shade all values to the right. Use a closed circle if the inequality is ≥ or ≤.
What is the distributive property in the context of inequalities?
It states a(b + c) = ab + ac. Apply it to inequalities exactly as with equations to remove parentheses before solving.
How is solving a two-step inequality different from solving a two-step equation?
The process is identical except for one rule: multiplying or dividing by a negative reverses the inequality symbol. Equations have no such directional sign.