Using Algebraic Symbols Flashcards

A student is saving for a new tablet that costs $350. They already have $80 saved. They earn $15 per week for chores. Which inequality represents the number of weeks, `w`, the student must work to have *at least* enough money to buy the tablet?

Answer: 80 + 15w ≥ 350

The student's total savings can be represented by their initial amount ($80) plus the amount they earn over time ($15 multiplied by the number of weeks, `w`), which is `80 + 15w`. The phrase 'at least' means the total must be greater than or equal to the goal of $350. Therefore, the correct inequality is `80 + 15w ≥ 350`.

The expression `100 - 8x` can be used to represent the amount of water, in gallons, remaining in a tank after `x` minutes. Which of the following statements best describes the meaning of the term `8x` in this context?

Answer: The total amount of water that has been drained from the tank after `x` minutes.

In the expression `100 - 8x`, the number 100 represents the initial quantity. The term `- 8x` represents the change over time. Since it is being subtracted, it signifies a decrease. The term `8x` itself represents the rate of decrease (8 gallons per minute) multiplied by the time (`x` minutes), which calculates the total amount of water drained after `x` minutes.

Which algebraic expression correctly represents the phrase 'five less than the product of a number, `n`, and the square of another number, `m`'?

Answer: nm² - 5

'The product of a number, `n`, and the square of another number, `m`' translates to `n * m²` or `nm²`. The phrase 'five less than' indicates that 5 should be subtracted from this product. Therefore, the correct expression is `nm² - 5`.

A coach buys `g` sports drinks for $2 each and `g` protein bars for $3 each for their team. The total cost can be represented by the expression `2g + 3g`. Which of the following expressions is NOT an equivalent way to represent the total cost?

Answer: 5g²

The original expression `2g + 3g` can be simplified by combining like terms to `5g`. Using the distributive property, `2g + 3g` is also equivalent to `g(2 + 3)` or `(2 + 3)g`. The expression `5g²` is not equivalent because it involves an exponent, which does not correctly model the total cost of the items.

A group of `f` friends wins a cash prize of `P` dollars. They agree to first donate $100 to a charity and then split the remaining amount equally among themselves. Which expression represents the amount of money each friend receives?

Answer: (P - 100) / f

The order of operations is crucial. First, the $100 donation is subtracted from the total prize money `P`, which is represented by `(P - 100)`. Then, this remaining amount is split equally among `f` friends, which means dividing by `f`. The parentheses are necessary to ensure the subtraction happens before the division.

Which verbal statement accurately describes the algebraic expression `(x + 4)³`?

Answer: The cube of the sum of a number and four.

The parentheses indicate that the operation inside, the sum of a number `x` and four (`x + 4`), must be performed first. The exponent `³` means that this entire sum is then cubed (raised to the third power). Therefore, the correct translation is 'The cube of the sum of a number and four.'