To solve an equation or inequality, the most straightforward method is to manipulate it until the variable is alone on one side and the constant is alone on the other. In other words, isolation.
Example
Whenever you solve an equation like 5r-2=10-r, you can check an answer by subbing it into the original equation. If we multiply or divide an inequality by a negative number, we must reverse the direction of the sign.
Strict and Non-Strict Inequalities
- Strict inequalities: the two sides cannot be equal.
- Non-Strict inequalities: the two sides may be equal.
Interval Notation
- (): values not included
- []: Values are included
- Negative infinity
- Positive infinity
- All real numbers
Examples
- TE[3,5) = 3²t>5
- TE(4+0)=>4
When taking a square root of both sides of an equation, don't forget negative values. For example, x²=a² [x=a].
Concept: Just as you can isolate the variable, in more complex equations, you can isolate an expression.
Substitution
One strategy to solve a system of equations is substitution. To solve one of the equations for one variable, then substitute it into the other equations. Usually, when we solve a 2-variable system, we write the answer as an ordered pair. If you have to assign multiple variables for a problem, use variables that are clearly related to what they represent.
Elimination
Elimination is another strategy to solve a system of equations. You combine the equations to produce new equations with fewer variables. When using elimination, look for the easiest variables to eliminate first. Elimination is the most effective for nonlinear equations. Don't just blindly eliminate variables!
By following these steps and understanding the rules for solving inequalities and systems of equations, you can successfully work through problems and improve your algebra skills. For additional practice, you can use solving inequalities algebra 1 worksheets and solving system of equations by elimination examples to enhance your understanding of the concepts.
