Free ATI TEAS 7 Mathematics guide
October 2026 - Variables & Equations Study Guide for nursing students in
Solving an equation means isolating the variable: do the same operation to both sides, and undo operations in reverse order. Then substitute your answer back into the original equation — a five-second check that catches almost every arithmetic slip.
The one rule
An equation is a balance. Anything you do to one side you must do to the other, and that single rule generates every technique.
- To remove something added, subtract it from both sides.
- To remove something subtracted, add it to both sides.
- To remove a multiplier, divide both sides by it.
- To remove a divisor, multiply both sides by it.
Undo in reverse order
To build 3x + 7, you multiply by 3 and then add 7. To take it apart, you undo in the opposite order: subtract 7 first, then divide by 3. Working through PEMDAS forwards when you should be reversing it is the most common structural error in this topic.
So for 3x + 7 = 22: subtract 7 from both sides to get 3x = 15, then divide both sides by 3 to get x = 5. Substituting back, 3(5) + 7 = 22, which confirms it.
A multi-step equation
Worked example
Solve 5(x - 2) = 3x + 8.
- Expand the bracket:
5x - 10 = 3x + 8. - Gather the variable on one side — subtract
3xfrom both:2x - 10 = 8. - Gather the numbers on the other — add 10 to both:
2x = 18. - Divide both sides by 2:
x = 9. - Check in the original:
5(9 - 2) = 35and3(9) + 8 = 35. They match.
x = 9. Expand first, collect variables on one side and numbers on the other, then divide.
Translating words into an equation
Word problems are the harder half of this topic, and the translation is mechanical once you know the vocabulary.
| Words | Symbol |
|---|---|
| sum, total, more than, increased by | + |
| difference, less than, decreased by, fewer | − |
| product, of, times, twice, per | × |
| quotient, per, divided into, ratio | ÷ |
| is, was, will be, results in, equals | = |
| a number, an unknown quantity, how many | the variable |
"Less than" reverses the order. "Five less than a number" is x - 5, not 5 - x. Written subtraction in English runs backwards from the arithmetic, and questions rely on it.
Inequalities, briefly
- Solve them exactly like equations, with one exception.
- Multiplying or dividing by a negative flips the sign. If
-2x > 6, thenx < -3. - Read the symbol carefully:
≤includes the endpoint,<does not.
Terms to know
- Variable
- A letter standing for an unknown value.
- Coefficient
- The number multiplying a variable. In
4x, the coefficient is 4. - Constant
- A number with no variable attached.
- Term
- A single part of an expression, separated by + or −.
- Like terms
- Terms with the same variable, which can be combined:
3x + 5x = 8x. - Isolate
- Get the variable alone on one side of the equation.
- Substitute
- Replace a variable with a value — how you check an answer.
What the TEAS asks most
The same core ideas appear in different wording. If you can answer these without stopping to think, you have what this section requires.
Solve for x.
Undo operations in reverse order, doing the same thing to both sides, then substitute back to check.
Which equation represents the situation described?
Translate phrase by phrase, and watch "less than", which reverses the subtraction.
If x = 4, what is the value of the expression?
Substitute, then follow the order of operations forwards.
Solve the inequality.
As for an equation, but flip the sign if you multiply or divide by a negative.
Key points
- Do the same operation to both sides — that is the whole method.
- Undo operations in reverse order: addition before multiplication.
- Collect variables on one side and constants on the other.
- Substitute your answer back into the original equation to check it.
- "Five less than a number" is
x - 5. Multiplying an inequality by a negative flips the sign.
Review quiz
4 questions on this topic, each with the reasoning worked through. Pick an answer to see how it went, or open the explanation straight away.
-
Question 1 of 4
Solve for x:
4x - 9 = 23.Show the answer and explanation
Correct answer: B.
x = 8Add 9 to both sides for
4x = 32, then divide by 4 to getx = 8, which checks out as4(8) - 9 = 23.x = 3.5comes from dividing before adding, 14 from subtracting 9 instead of adding it, and 128 from multiplying by 4 rather than dividing. -
Question 2 of 4
Which equation represents "seven less than three times a number is twenty"?
Show the answer and explanation
Correct answer: B.
3n - 7 = 20"Three times a number" is
3n, and "seven less than" that subtracts 7 from it, giving3n - 7 = 20. The first choice reverses the subtraction — the classic trap — and the third subtracts 7 before multiplying, which describes something different again. -
Question 3 of 4
Solve
2(x + 3) = x + 11.Show the answer and explanation
Correct answer: A.
x = 5Expanding gives
2x + 6 = x + 11; subtractingxand then 6 from both sides leavesx = 5. Substituting confirms it:2(8) = 16on the left and16on the right. The wrong choices come from failing to distribute the 2 across both terms in the bracket. -
Question 4 of 4
Solve the inequality
-3x ≥ 12.Show the answer and explanation
Correct answer: B.
x ≤ -4Dividing both sides by −3 gives
x ≤ -4, because dividing by a negative flips the inequality. Testing a value settles it:x = -5gives15 ≥ 12, which holds, whilex = 0gives0 ≥ 12, which does not — so the solution set must be the values at or below −4.
Put this TEAS topic into practice
Build a set on variables and equations and use the rationale on every question to reinforce what you just reviewed.
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