Chapter 6: Equations & Inequalities
An equation is a mathematical statement that two expressions are equal. Solving an equation means finding the value(s) of the variable that make the statement true. In this chapter we build from simple one-step equations all the way to equations with variables on both sides, and then extend these ideas to inequalities — statements about which values are larger or smaller than a given quantity.
6.1 Equations and Solutions
Definition — Equation
An equation is a statement of the form $\text{expression}_1 = \text{expression}_2$. A solution is a value of the variable that makes the equation true.
Example: In $x + 3 = 10$, the solution is $x = 7$ because $7 + 3 = 10$. ✓
Properties of Equality (The Golden Rule of Equations)
Whatever you do to one side of an equation, you must do the same to the other side.
- Addition Property: If $a = b$, then $a + c = b + c$.
- Subtraction Property: If $a = b$, then $a - c = b - c$.
- Multiplication Property: If $a = b$, then $ac = bc$.
- Division Property: If $a = b$ and $c \neq 0$, then $\dfrac{a}{c} = \dfrac{b}{c}$.
6.2 One-Step Equations
A one-step equation requires exactly one operation to isolate the variable. You "undo" the operation by applying its inverse.
| Type | Example | Inverse Operation | Solution |
|---|---|---|---|
| Addition | $x + 9 = 15$ | Subtract 9 | $x = 6$ |
| Subtraction | $y - 4 = 11$ | Add 4 | $y = 15$ |
| Multiplication | $3n = 21$ | Divide by 3 | $n = 7$ |
| Division | $\dfrac{m}{5} = 8$ | Multiply by 5 | $m = 40$ |
Example 1 — One-Step with Fractions
Solve $\dfrac{3}{4}x = 9$.
Solution: Multiply both sides by the reciprocal $\dfrac{4}{3}$.
$$\frac{4}{3} \cdot \frac{3}{4}x = 9 \cdot \frac{4}{3} \implies x = 12$$Check: $\dfrac{3}{4}(12) = 9$ ✓
6.3 Two-Step Equations
Two-step equations require two inverse operations. The standard strategy is to undo addition or subtraction first, then undo multiplication or division.
Strategy for Two-Step Equations
- Add or subtract a constant from both sides to move the constant term to the right.
- Multiply or divide both sides to isolate the variable.
- Always check your answer by substituting back.
Example 2 — Two-Step Equation
Solve $3x - 7 = 14$.
Check: $3(7) - 7 = 21 - 7 = 14$ ✓
Example 3 — Negative Coefficient
Solve $-4y + 5 = -11$.
Check: $-4(4) + 5 = -16 + 5 = -11$ ✓
6.4 Multi-Step Equations
Multi-step equations may require distributing, combining like terms, and then solving. Always simplify each side completely before moving terms across the equals sign.
Example 4 — Distribute Then Solve
Solve $2(3x - 1) + 4 = 20$.
Check: $2(9-1)+4 = 16+4 = 20$ ✓
6.5 Variables on Both Sides
When variables appear on both sides, collect all variable terms on one side and all constant terms on the other.
Example 5 — Variables on Both Sides
Solve $5x + 3 = 2x + 18$.
Check: $5(5)+3 = 28$; $2(5)+18 = 28$ ✓
Special Cases
- No solution: If variables cancel and the result is a false statement (e.g., $3 = 7$), there is no solution.
- Infinitely many solutions: If variables cancel and the result is a true statement (e.g., $0 = 0$), every real number is a solution (identity).
6.6 Introduction to Inequalities
Inequality Symbols
| Symbol | Meaning | Example |
|---|---|---|
| $<$ | less than | $x < 4$ |
| $>$ | greater than | $y > -2$ |
| $\leq$ | less than or equal to | $n \leq 10$ |
| $\geq$ | greater than or equal to | $m \geq 0$ |
An inequality has infinitely many solutions — a whole range of numbers. We graph this range on a number line using:
- An open circle (○) for strict inequalities ($<$ or $>$) — the endpoint is not included.
- A closed circle (●) for inclusive inequalities ($\leq$ or $\geq$) — the endpoint is included.
- An arrow pointing in the direction of the solution set.
Graphing on the Number Line
Use the interactive Desmos tool below to explore inequality graphs:
Examples of number-line representations:
6.7 Solving One-Step and Two-Step Inequalities
Inequalities are solved just like equations, with one critical exception:
The Flip Rule
When you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality symbol.
$$-2x < 8 \implies x > -4 \quad \text{(divided by } -2\text{, flipped } < \text{ to } >\text{)}$$Example 6 — One-Step Inequality
Solve and graph $x + 5 \leq 9$.
Graph: Closed circle at 4, arrow pointing left.
Solution set: All real numbers less than or equal to 4; interval notation: $(-\infty, 4]$.
Example 7 — Two-Step Inequality with Flip
Solve $-3n + 6 > 15$.
Graph: Open circle at $-3$, arrow pointing left.
Check with $n = -4$: $-3(-4)+6 = 18 > 15$ ✓
Practice Problems
Practice — Chapter 6
- Solve: $x - 13 = -5$.
- Solve: $\dfrac{x}{-4} = 6$.
- Solve: $5y + 2 = 27$.
- Solve: $4(2x - 3) = 20$.
- Solve: $7k - 4 = 3k + 12$.
- Determine whether $x = -2$ is a solution of $3x + 11 = 5$.
- Solve and graph: $2x - 1 > 7$.
- Solve: $-5m + 4 \leq 19$. State the solution set using interval notation.
- A ticket costs $\$12$ and you have at most $\$60$. Write and solve an inequality for the number of tickets $t$ you can buy.
Show Answers
- $x = 8$
- $x = -24$
- $y = 5$
- $8x - 12 = 20 \Rightarrow 8x = 32 \Rightarrow x = 4$
- $4k = 16 \Rightarrow k = 4$
- $3(-2)+11 = 5$ ✓ Yes, $x=-2$ is a solution.
- $2x > 8 \Rightarrow x > 4$; open circle at 4, arrow right.
- $-5m \leq 15 \Rightarrow m \geq -3$; interval: $[-3, \infty)$.
- $12t \leq 60 \Rightarrow t \leq 5$; at most 5 tickets.
Chapter Summary
- An equation states two expressions are equal; a solution is the value that makes it true.
- Use inverse operations and the properties of equality to isolate the variable.
- For multi-step equations: distribute, combine like terms, then solve.
- When variables appear on both sides, move them to one side first.
- An inequality has a range of solutions; graph with open (strict) or closed (inclusive) circles and arrows.
- The flip rule: reversing inequality direction is required when multiplying or dividing by a negative number.