2.2 Algebra and Equations
Key Takeaways
- Simplify expressions before solving equations.
- Use inverse operations consistently on both sides.
- Inequality symbols reverse when multiplying or dividing by a negative number.
- Check answers by substitution, especially after clearing denominators or squaring.
The Algebra Order
Most algebra misses come from doing correct operations in the wrong order. Start by simplifying each side: distribute, combine like terms, reduce fractions where allowed, and only then isolate the variable.
Equation Workflow
- Clear grouping: remove parentheses by distributing.
- Combine terms: put like terms together.
- Collect variables: move variable terms to one side.
- Undo operations: use inverse operations in reverse order.
- Check: substitute the answer into the original equation.
Special Cases
| Case | Extra step |
|---|---|
| Inequality | Flip sign after negative multiplication or division |
| Rational equation | Reject denominator-zero answers |
| Radical equation | Check for extraneous solutions |
| Quadratic equation | Set equal to zero first |
| System | Use substitution or elimination |
ALEKS often tests whether you know the difference between simplifying an expression and solving an equation. An expression has no equals sign; your job is to rewrite it. An equation has a condition; your job is to find values that make it true.
Practice Focus
Work several versions of the same equation type until the procedure is automatic. For example, solve one equation with parentheses, one with fractions, one with variables on both sides, and one inequality that requires a sign flip. This variety keeps you from memorizing a single pattern and prepares you for the adaptive format.
When solving -2x > 10, what happens when both sides are divided by -2?