Study guide: Solve Equations and Represent Constraints
Solve additive and multiplicative equations, interpret inequalities, and model dependent variables.
Ideas to explain in your own words
- An equation states two expressions have the same value.
- A positive coefficient times an unknown can be undone by dividing both sides by that coefficient.
- A strict inequality such as x>c excludes its boundary c.
- A relationship can define one quantity from another.
Useful vocabulary
- solution
- An input making an equation or inequality true.
- dependent variable
- A quantity whose modeled value depends on another variable.
Watch your reasoning
- Check the result in the original equation, not only the last simplified line.
- A solution need not be a whole number unless the model restricts it.
- Real-number and whole-number domains give different allowable solution sets.
- A fictional rule does not certify how a real device behaves.
A worked review example
Solve x+4.5=12. Subtract 4.5 from both sides: x=7.5. Checking 7.5+4.5=12 verifies the original sentence. Subtracting from only one side would create a different relationship rather than a justified equivalent step. Apply each balancing operation to both sides. Check the result in the original equation, not only the last simplified line. Draw equal balance bars and cross off the same known amount on both sides.
Make a practice plan
- Solve an Additive Equation by Preserving Equality: Solve a+3/4=2. Expected: a=5/4.
- Solve a Multiplicative Equation by Division: Solve 4x=30. Expected: x=7.5.
- Represent a Strict Inequality and Its Boundary: Represent real t<2. Expected: open circle at two, shading left; 2 excluded.
- Connect a Dependent Variable to a Table and Rule: For y=3x+2 at x=0,1,3, expected outputs are 2,5,11.