# SAT II Math II : Single-Variable Algebra

## Example Questions

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### Example Question #1 : Single Variable Algebra

Which of the following phrases can be written as the algebraic expression  ?

The absolute value of the product of negative eight and a number

The absolute value of the difference of eight and a number

Eight subtracted from the absolute value of a number

Eight decreased by the absolute value of a number

The absolute value of the difference of a number and eight

The absolute value of the difference of eight and a number

Explanation:

is the absolute value of , which is the difference of eight and a number. Therefore,  is "the absolute value of the difference of eight and a number."

### Example Question #2 : Single Variable Algebra

Which of the following phrases can be written as the algebraic expression  ?

Seven decreased by the absolute value of a number

The absolute value of the difference of seven and a number

Seven decreased by the opposite of a number

The opposite of a number decreased by seven

The opposite of the difference of seven and a number

Seven decreased by the opposite of a number

Explanation:

is seven decreased by , which is the opposite of a number; therefore,  is "seven decreased by the opposite of a number."

### Example Question #3 : Single Variable Algebra

Which of the following phrases can be represented by the algebraic expression  ?

Nine less than by the multiplicative inverse of a number

Nine decreased by the multiplicative inverse of a number

The multiplicative inverse of the difference of a number and nine

The multiplicative inverse of the difference of nine and a number

One divided into the difference of nine and a number

The multiplicative inverse of the difference of nine and a number

Explanation:

is the multiplicative inverse of , which is the difference of nine and a number. Therefore,  is "the multiplicative inverse of the difference of nine and a number".

### Example Question #4 : Single Variable Algebra

Which of the following phrases can be represented by the algebraic expression

Ten less than three times the square root of a number

The cube root of the difference of a number and ten

Ten decreased by three times the square root of a number

Ten less than the cube root of a number

Ten decreased by the cube root of a number

Ten less than the cube root of a number

Explanation:

is ten less than , which is the cube root of a number; therefore,  is "ten less than the cube root of a number".

### Example Question #5 : Single Variable Algebra

Which of the following phrases can be represented by the algebraic expression

The square root of the difference of a number and twenty

Twenty less than the square root of a number

The square root of the difference of twenty and a number

Negative twenty multiplied by the square root of a number

Twenty decreased by the square root of a number

Twenty decreased by the square root of a number

Explanation:

is twenty decreased by , which is the square root of a number, so  is "twenty decreased by the square root of a number".

### Example Question #2 : Single Variable Algebra

Adult tickets to the zoo sell for ; child tickets sell for . On a given day, the zoo sold  tickets and raised  in admissions. How many adult tickets were sold?

Explanation:

Let  be the number of adult tickets sold. Then the number of child tickets sold is .

The amount of money raised from adult tickets is ; the amount of money raised from child tickets is . The sum of these money amounts is , so the amount of money raised can be defined by the following equation:

To find the number of adult tickets sold, solve for :

### Example Question #1 : Single Variable Algebra

Sarah sells lemonade at the concession stands.  She charges fifty cents per cup of lemonade, and twenty five cents for refills.  What is the equation that represents the total that she will make from the lemonade stand using the variables cups  and refills ?

Explanation:

Sarah charges fifty cents per cup of lemonade:

Sarah charges twenty five cents for refills:

Set up the equation by adding the totals.

### Example Question #2 : Single Variable Algebra

Solve for :

Explanation:

To solve for , you need to isolate it to one side of the equation. You can subtract the  from the right to the left. Then you can add the 6 from the right to the left:

Next, you can factor out this quadratic equation to solve for . You need to determine which factors of 8 add up to negative 6:

Finally, you set each binomial equal to 0 and solve for :

### Example Question #1 : Single Variable Algebra

Give the solution set of the equation  .

Explanation:

Use the quadratic formula with :

### Example Question #1 : Single Variable Algebra

A large tub has two faucets. The hot water faucet, if turned all the way up, can fill the tub in 15 minutes; the cold water faucet can do the same in 9 minutes. Which of the following responses is closest to the time it takes to fill the tub if both faucets are turned all the way up?

Explanation:

Work problems can be solved by looking at them as rate problems.

The hot faucet can fill up the tub at a rate of 15 minutes per tub, or  tub per minute. The cold faucet, similarly, can fill up the tub at a rate of 9 minutes per tub, or  tub per minute.

Suppose the tub fills up in  minutes. Then, at the end of this time, the hot faucet has filled up  tub, and the cold faucet has filled up  tub, for a total of one tub. We can set up this equation and solve for :

Of the five choices, 6 minutes comes closest.

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