What the question looks like
Pre-algebra questions are the ones with a letter in them. You might be asked to solve for x, to evaluate an expression when you are told what the letters equal, or to find the output of a simple rule like "if y = 3x − 2, what is y when x = 4?"
The topics include evaluating expressions, one- and two-step equations, simple inequalities, integers and number lines, patterns and simple functions, and basic proportions. A few items dress the algebra up as a short sentence, such as "Five more than twice a number is 19. What is the number?"
Everything stays at the pre-algebra level. You will never see a squared variable to solve for, two equations at once, or anything you would meet later in Algebra I.
What you need to know
- Whatever you do to one side of an equation, do to the other side.
- To solve a two-step equation, undo addition or subtraction first, then undo multiplication or division.
- To evaluate an expression, replace each letter with its value, put negative values in parentheses, then follow order of operations.
- Subtracting a negative is the same as adding: 6 − (−6) = 12.
- Multiplying two negatives gives a positive. Multiplying a negative and a positive gives a negative.
- On a number line, numbers get larger to the right and smaller to the left, so −2 is greater than −7.
- An inequality is solved like an equation, but flip the sign if you multiply or divide both sides by a negative number.
- A function rule such as y = 5x + 3 tells you exactly what to do with x: multiply by 5, then add 3.
- A proportion is two equal ratios. Cross-multiply to solve for the missing value.
- "Twice a number" means 2n. "Five more than" means + 5. "Is" means equals.
How to solve it
- Identify what the question is asking for. Are you solving for a letter, plugging in a value, or extending a pattern?
- Write the equation or expression down exactly, including any negative signs. Do not try to hold it in your head.
- Work one step at a time, showing each step in the margin. For equations, undo addition or subtraction before multiplication or division.
- Check your answer by putting it back into the original equation. This takes five seconds and catches most errors.
- Match your result to a choice, and if it is not there, go back to the first step where a sign could have flipped.
Worked examples
Example 1
If 4x − 7 = 21, what is the value of x?
- A.3.5
- B.5.25
- C.7
- D.28
Show answer
Answer: C: 7
Undo the subtraction first by adding 7 to both sides: 4x = 28. Then divide both sides by 4: x = 7. Check: 4 × 7 − 7 = 28 − 7 = 21. Choice A, 3.5, comes from subtracting 7 instead of adding it and then dividing 14 by 4, and choice D, 28, is what you get if you forget to divide.
Example 2
If a = 3 and b = −2, what is the value of 2a − 3b?
- A.−12
- B.0
- C.6
- D.12
Show answer
Answer: D: 12
Substitute carefully: 2a = 2 × 3 = 6, and 3b = 3 × (−2) = −6. Then 6 − (−6) = 6 + 6 = 12. Choice B, 0, is the most tempting mistake, because it comes from treating b as positive 2 and computing 6 − 6.
Example 3
If y = 5x + 3, what is y when x = 6?
- A.14
- B.30
- C.33
- D.45
Show answer
Answer: C: 33
Replace x with 6 and follow the rule: 5 × 6 = 30, then 30 + 3 = 33. Choice D, 45, comes from adding 6 + 3 first and then multiplying by 5, which breaks order of operations. Choice B, 30, forgets the final step.
Example 4
A bread recipe uses 3 cups of flour for every 2 cups of milk. How many cups of flour are needed for 8 cups of milk?
- A.5
- B.9
- C.12
- D.16
Show answer
Answer: C: 12
Set up the proportion 3/2 = f/8. Cross-multiply: 2 × f = 3 × 8 = 24, so f = 12. Another way to see it is that 8 cups of milk is 4 times as much as 2 cups, so you need 4 times as much flour, and 3 × 4 = 12. Choice B, 9, comes from adding 6 to the flour because 6 was added to the milk, but proportions scale by multiplying, not adding.
Traps to watch for
- Lost negative signs. A dropped minus sign turns a correct answer into a wrong one, and the wrong answer is always sitting in the choices. Put every negative value in parentheses when you substitute.
- Undoing operations in the wrong order. In 4x − 7 = 21, deal with the − 7 before the 4. Doing the division first gives a fraction and a wrong answer.
- Adding when you should multiply. Proportions and patterns scale by a factor. If one quantity quadruples, the other quadruples too.
- Answering for the wrong letter. Some questions solve for x and then ask for 2x or x + 1. Read the last sentence twice before you bubble.
Pacing
At about 42 seconds per question, a two-step equation should take you 20 to 30 seconds including the check. Handle the ones you recognize first and come back to anything that needs more than a minute of thought. There is no guessing penalty, so pick a choice for every question, even the ones you skipped.
Practice
See how your pre-algebra speed compares to the real clock by taking our free diagnostic.
Shaky on the skill itself? Start with the lessons on elementary algebra, integers and order of operations, and exponents and roots, then come back to the question format.