Practice Algebra as Language in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

The perspective that algebra is a formal language with syntax and grammar for expressing mathematical ideas and relationships precisely.

Just as English has grammar, algebra has rules for combining symbols meaningfully.

Showing a random 20 of 50 problems.

Example 1

medium
Express in algebra: 'The product of two consecutive integers is 12 more than twice their sum.'

Example 2

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Express: 'The square of the sum of aa and bb' vs 'The sum of the squares of aa and bb'. Are they equal?

Example 3

easy
Translate 'eight less than three times a number nn' into an algebraic expression.

Example 4

easy
Translate 'five more than a number xx' into algebra.

Example 5

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Read f:R→R, f(x)=x2f: \mathbb{R} \to \mathbb{R},\ f(x) = x^2 in plain English.

Example 6

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Translate Pythagoras's relation 'the square on the hypotenuse equals the sum of the squares on the other two sides' with legs a,ba,b and hypotenuse cc.

Example 7

challenge
Express 'a function ff is injective' in formal logic.

Example 8

easy
Translate 'half of a number decreased by 4' into algebra.

Example 9

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A taxi charges $3 plus $2 per mile. Write the cost CC for mm miles as an equation.

Example 10

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A student writes 'x=3=yx = 3 = y' to mean x=3x = 3 and y=3y = 3. Why is this a grammar violation?

Example 11

easy
Translate 'the sum of three consecutive even integers' starting from nn into an expression.

Example 12

hard
Read ∑k=1nk=n(n+1)2\sum_{k=1}^{n} k = \dfrac{n(n+1)}{2} as a sentence.

Example 13

easy
Translate 'a number is positive' into an inequality with variable xx.

Example 14

easy
Translate 'xx is between 2 and 8 inclusive' into a compound inequality.

Example 15

easy
Read the expression a+bc\frac{a+b}{c} in words.

Example 16

medium
Read (x+1)2−(x−1)2(x+1)^2 - (x-1)^2 in words, then simplify.

Example 17

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Translate the constraint 'a number is no more than twice another, and they sum to 12' into a system using x,yx,y.

Example 18

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Is the expression '3+×43 + \times 4' grammatically valid algebra?

Example 19

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Write an equation: 'A number decreased by 7 equals 15.'

Example 20

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Read ∣x∣≤3|x| \le 3 in words and express it as a compound inequality without absolute value.
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