Distributing Negative Signs and Handling Parentheses with Negatives
Distributing negative signs correctly across parentheses is a critical algebraic technique in Grade 11 enVision Algebra 1 (Chapter 1: Solving Equations and Inequalities). The rule is: −(a + b) = −a − b and −(a − b) = −a + b. When a negative coefficient precedes parentheses, every term inside changes sign. A common error is only negating the first term. Understanding that subtracting a positive equals adding a negative, and subtracting a negative equals adding a positive, prevents systematic sign mistakes.
Key Concepts
When distributing a negative sign across parentheses: $ (a + b) = a b$ and $ (a b) = a + b$.
When multiplying by a negative coefficient: $ c(a + b) = ca cb$ and $ c(a b) = ca + cb$.
Common Questions
What happens when you distribute a negative sign across parentheses?
Every term inside the parentheses changes sign: −(a + b) = −a − b, and −(a − b) = −a + b.
Why do both terms change sign, not just the first?
Distributing a negative sign is equivalent to multiplying by −1, which must be applied to every term inside the parentheses by the distributive property.
What is the most common error when distributing negatives?
Only changing the sign of the first term inside the parentheses and leaving the remaining terms unchanged.
How do you handle −3(x − 5)?
Distribute −3 to each term: −3 × x = −3x and −3 × (−5) = +15, giving −3x + 15.
How does subtracting a negative work with parentheses?
Subtracting a negative is the same as adding a positive: a − (−b) = a + b. The two negatives combine to produce a positive.
How do you simplify 4 − (2x − 7)?
Distribute the negative: 4 − 2x + 7 = 11 − 2x. Remember both terms inside change sign.