November 2026 - Solving 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.
Try 4 short questions throughout this guide. Check each answer to learn from the explanation.
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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.
Question 1 of 4
Solve for x: 4x - 9 = 23.
Choose one answer, then check it.
View the Answer and Explanation
Correct answer: B. x = 8
Add 9 to both sides for 4x = 32, then divide by 4 to get x = 8, which checks out as 4(8) - 9 = 23. x = 3.5 comes from dividing before adding, 14 from subtracting 9 instead of adding it, and 128 from multiplying by 4 rather than dividing.
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 3x from 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) = 35 and 3(9) + 8 = 35. They match.
x = 9. Expand first, collect variables on one side and numbers on the other, then divide.
Question 2 of 4
Solve 2(x + 3) = x + 11.
Choose one answer, then check it.
View the Answer and Explanation
Correct answer: A. x = 5
Expanding gives 2x + 6 = x + 11; subtracting x and then 6 from both sides leaves x = 5. Substituting confirms it: 2(8) = 16 on the left and 16 on the right. The wrong choices come from failing to distribute the 2 across both terms in the bracket.
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, then x < -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 HESI 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.
Question 3 of 4
Which equation represents "seven less than three times a number is twenty"?
Choose one answer, then check it.
View 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, giving 3n - 7 = 20. The first choice reverses the subtraction — the classic trap — and the third subtracts 7 before multiplying, which describes something different again.
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.
Question 4 of 4
Solve the inequality -3x ≥ 12.
Choose one answer, then check it.
View the Answer and Explanation
Correct answer: B. x ≤ -4
Dividing both sides by −3 gives x ≤ -4, because dividing by a negative flips the inequality. Testing a value settles it: x = -5 gives 15 ≥ 12, which holds, while x = 0 gives 0 ≥ 12, which does not — so the solution set must be the values at or below −4.
HESI · Solving equations
Knowledge Check Results
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