Algebraic expressions and solving equations — ISEE Upper practice questions

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Practice 168 Algebraic expressions and solving equations questions in the app

What this topic is

On the ISEE Upper Quantitative Reasoning section, Algebraic expressions and solving equations covers combining like terms, simplifying polynomials, and turning word problems into equations.

A student must translate a relationship between the digits of a two-digit number into algebra, add or subtract polynomials, and apply a constraint such as x greater than one when comparing two expressions. Items often pit Column A against Column B, with choices that one column is greater, that they are equal, or that the relationship cannot be determined, alongside ordinary multiple-choice word problems.

Traps include dropped signs, mixed-up place values, and treating an inequality as a single value.

Sample questions

Pick an answer to see whether it is right — nothing is saved, and nothing needs an account.

Question 1Mid

x>1

Column A	Column B
$x+\dfrac{1}{x}$	2

The quantity in Column A is greater✓ CorrectThe quantity in Column B is greater✗ Not this oneThe two quantities are equal✗ Not this oneThe relationship cannot be determined from the information given✗ Not this one

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Answer: A — The quantity in Column A is greater

For $x>1$ , $(x-1)^2>0$ , so $x^2+1>2x$ . Dividing by positive $x$ gives $x+\dfrac{1}{x}>2$ .

Question 2Mid

A two-digit number has a tens digit that is twice its ones digit. When the digits are reversed, the new number and the original number have a sum of 99. What is the original number?

36✗ Not this one42✗ Not this one63✓ Correct84✗ Not this one

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Answer: C — 63

The original number is $20a+a=21a$ , and the reversed number is $10a+2a=12a$ . Their sum is $33a=99$ , so $a=3$ and the original number is 63.

Question 3Easier

Column A	Column B
$(3x^2-4x+6)+(2x^2+7x-9)$	$5x^2+3x-3$

The quantity in Column A is greater✗ Not this oneThe quantity in Column B is greater✗ Not this oneThe two quantities are equal✓ CorrectThe relationship cannot be determined from the information given✗ Not this one

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Answer: C — The two quantities are equal

Combine like terms: $3x^2+2x^2=5x^2$ , $-4x+7x=3x$ , and $6-9=-3$ . The first expression is identical to $5x^2+3x-3$ , so the two quantities are equal.

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Practice 168 Algebraic expressions and solving equations questions in the app

Algebraic expressions and solving equations — ISEE Upper practice questions

What this topic is

Sample questions

More ISEE Upper topics

Quantitative Reasoning