Special Case: Division by -1
Division by −1 is a special case in Grade 7 algebra in Big Ideas Math, Course 2 that appears when solving equations or inequalities with a −1 coefficient on the variable. Dividing both sides by −1 is equivalent to multiplying by −1, which negates both sides and changes the sign of every term. For example, −x = 5 divided by −1 gives x = −5. In inequalities, dividing by −1 also reverses the inequality symbol: −x > 3 becomes x < −3. Recognizing this case prevents students from leaving −x as the solution and ensures the variable has a positive coefficient in the final answer.
Key Concepts
Dividing any integer by $ 1$ gives its opposite. $$a \div ( 1) = a$$.
Common Questions
What does dividing both sides of an equation by −1 do?
It negates both sides, changing every term's sign. −x = 5 becomes x = −5 after dividing both sides by −1.
How do you solve −x = 5?
Divide both sides by −1: (−x)/(−1) = 5/(−1), giving x = −5.
Does dividing by −1 reverse an inequality symbol?
Yes. Dividing any inequality by a negative number reverses the symbol. −x > 3 becomes x < −3.
Why can't you leave −x as your final answer?
A solution should express the variable with a positive coefficient. −x = something means x equals the negative of that value; you must complete the division.
Is dividing by −1 the same as multiplying by −1?
Yes—multiplying and dividing by −1 are equivalent operations. Both negate both sides of the equation or inequality.
How do you recognize when division by −1 is needed?
When the variable term is negative after isolating it, such as −x = k or −1·x = k, you need to divide (or multiply) by −1 to solve for x.