7.1 Expressions, Equations, and Inequalities
Key Takeaways
- The GED math test expects you to simplify, evaluate, and write algebraic expressions with rational coefficients, not just recognize symbols.
- Linear equations often require distribution, combining like terms, and one inverse operation at a time.
- Inequality symbols reverse only when multiplying or dividing both sides by a negative number.
- Real-world GED algebra questions usually start with a phrase that must become an expression, equation, or inequality before calculation begins.
- Checking an answer by substitution is one of the fastest ways to catch sign errors on test day.
GED Focus
Algebra on the GED Mathematical Reasoning test is practical. The official assessment targets include writing, evaluating, and computing with expressions; solving linear equations with rational coefficients; and solving or graphing one-variable inequalities. That means you should be ready to turn a situation into symbols, simplify without losing signs, and decide whether the final answer makes sense.
Expressions Are Instructions
An expression has no equals sign. It can be simplified or evaluated, but not solved. For example, 3(2x - 5) + 4x is an expression. Use the distributive property first: 6x - 15 + 4x. Then combine like terms: 10x - 15.
| Task | Example | What To Do |
|---|---|---|
| Simplify | 5x + 2 - 3x | Combine like terms: 2x + 2 |
| Evaluate | 4a - 7 when a = 6 | Substitute: 24 - 7 = 17 |
| Translate | 8 more than twice n | Write 2n + 8 |
| Equation | Twice n is 18 | Write 2n = 18 |
| Inequality | At least 18 | Write x >= 18 |
Worked Example: Solve A Linear Equation
Solve 4(x - 3) + 7 = 2x + 11.
- Distribute: 4x - 12 + 7 = 2x + 11.
- Combine like terms: 4x - 5 = 2x + 11.
- Subtract 2x from both sides: 2x - 5 = 11.
- Add 5: 2x = 16.
- Divide by 2: x = 8.
Check by substitution: 4(8 - 3) + 7 = 27, and 2(8) + 11 = 27. The answer works.
Inequalities Need Direction
An inequality compares values instead of making them equal. Solve it like an equation, except the symbol reverses when multiplying or dividing both sides by a negative number.
Solve -3x + 5 < 20.
Subtract 5: -3x < 15. Divide by -3 and reverse the symbol: x > -5. On a number line, use an open circle at -5 and shade to the right because values greater than -5 are solutions.
GED Strategy
Read the wording before touching the calculator. Words such as "more than," "less than," "at most," and "at least" decide the structure.
- "At least" means greater than or equal to.
- "No more than" means less than or equal to.
- "The sum of" usually signals addition.
- "The product of" signals multiplication.
- "Difference" order matters: "5 less than x" is x - 5.
Fast Practice Routine
Build fluency with short mixed drills: one simplification, one substitution, one equation, and one inequality. Say the reason for each step out loud while practicing, especially when dividing by a negative. On the actual GED, you may see drag-and-drop, fill-in-the-blank, or multiple-choice algebra. The format changes, but the algebra rules do not. If answer choices are available, plug the most reasonable choice back into the original sentence before committing. If the problem includes decimals or fractions, clear them only when you can do so on both sides of the equation.
Otherwise, use the calculator carefully and keep the exact inequality or equation written on your scratch pad.
Most GED algebra misses come from one of three habits: distributing only the first term, dropping a negative sign, or reversing a phrase such as "less than." Keep your work vertical, line up equal signs or inequality signs, and check one answer choice by plugging it into the original problem.
Simplify 3(2x - 4) + 5x.
A GED question says, "A repair must cost no more than $180. The company charges $45 plus $30 per hour." Which inequality represents the number of hours h?