Why this chapter matters
Linear equations describe break-even points, missing measurements, rates, and many other situations where one unknown must be found.
What you will learn
- Solve multi-step equations and verify solutions by substitution.
- Recognize equations with one, none, or infinitely many solutions.
- Solve and graph inequalities while handling negative multiplication correctly.
Understand the core ideas
Solving a linear equation means isolating the unknown while preserving balance. Think of the equals sign as a scale: whatever operation you do on one side must also be done on the other. A reliable sequence is simplify each side first, move variable terms together, move constants together, then divide by the variable coefficient. Always check by substitution, because a single arithmetic slip can produce a value that looks reasonable but does not satisfy the original statement.
Inequalities follow almost the same process, with one key difference: multiplying or dividing by a negative reverses the direction symbol. For example, if -2x < 10, dividing by -2 gives x > -5, not x < -5. When equations simplify to a true statement like 4 = 4, there are infinitely many solutions. If they simplify to a false statement like 4 = 9, there is no solution. Naming these cases clearly is part of solving, not an extra step.
Key terms
- inverse operation
- An operation that undoes another, such as subtraction undoing addition.
- solution set
- All values that make an equation or inequality true.
- identity
- An equation true for every allowed value of the variable.
- contradiction
- An equation that is never true, so it has no solution.
Solve with variables on both sides
Solve 5(2x - 1) - 3 = 4x + 9.
- Distribute and simplify the left side: 5(2x - 1) - 3 = 10x - 5 - 3 = 10x - 8.
- Set 10x - 8 = 4x + 9 and subtract 4x from both sides to get 6x - 8 = 9.
- Add 8 to both sides: 6x = 17.
- Divide by 6: x = 17/6.
- Check by substitution in the original equation to confirm both sides are equal.
A common misconception
Claim: When solving inequalities, the sign always stays the same because you did the same steps on both sides.
Correction: The sign flips whenever you multiply or divide by a negative number. That rule is required to keep the inequality true.
Lessons in this chapter
- Solving linear equationsUse inverse operations and preserve equality at every step. Read the full guide →
- Variables on both sidesCollect variable terms and diagnose identities or contradictions.
- Solving inequalitiesReverse the inequality when multiplying or dividing by a negative value. Read the full guide →
- Modeling constraintsTranslate limits such as budgets and capacities into inequalities.
Study task
Chapter checkpoint
Solve 4(x - 2) + 3 = 2x + 9.
Expand to 4x - 5 = 2x + 9, subtract 2x, add 5, and get 2x = 14, so x = 7.