Decoding and Translating

Question 1

Decode this:

x - 11 | = 7

Answer 1

The absolute value of x minus 11 is equal to 7.

Logically, this means either…

• x* - 11 = 7 or
• x* - 11 = -7

So, x = 18 or

x = 4

Question 2

“…the first 15 square values…”

Answer 2

1, 4, 9, 16, 25, 36, 49, 64, 81,100,121,144,169,196, 225

Knowing your squares is a time saver. These represent the first 15 negative values squared as well.

Question 3

the product of the first three odd prime numbers

Answer 3

3 x 5 x 7

105

Question 4

the sum of a number and 11

Answer 4

n + 11

Question 5

the sum of a number and minus 7

Answer 5

n + (-7) or n - 7

Question 6

number increased by 100

Answer 6

n + 100

Question 7

number is increased by 25%

Answer 7

n + (n x .25) or

n + n / 4 or

n x (5/4) or

5n / 4

Question 8

2 greater than a number

Answer 8

n + 2

“Greater than” in this context means “more than”. Do not confuse it with “Is greater than” which means “>”.

Question 9

number is decreased by 14

Answer 9

n - 14

Question 10

5 less than a number

Answer 10

n - 5

Question 11

Decode:

…absolute value of a number…

Answer 11

|a | = a

if a is greater than or equal to zero.

|a| = -a

if a is less than zero.

Question 12

…the absolute value of the sum of a and b…

Answer 12

|a + b|

Question 13

…the sum of the absolute values of a and b…

Answer 13

|a| + |b|

Note: “…the product of the absolute value of a and b…” won’t be asked on the SAT. The result is identical to the “the product of the absolute values of a and b”.

Question 14

“…the sum of the first 8 prime numbers divided by the absolute value of the difference between the 5th and the 6th square values…”

Answer 14

2 + 3 + 5 + 7 + 11 + 13 + 17 + 19

77 / |25 - 36| = 7

Question 15

(a + b)(a - b)

Answer 15

(a + b)(a - b) = a² - b²

Question 16

a² - b²

Answer 16

a² - b² = (a + b)(a - b)

Question 17

(a + b)²

Answer 17

a²+ 2ab + b²

Question 18

(a - b)²

Answer 18

a²- 2ab + b²

Question 19

…what is n in terms of w and c…

Answer 19

n = some combination of w and c

To “solve or express in terms” means to isolate one variable on one side of the equation, leaving the other variables on the other side of the equation.

Question 20

…the two mixtures combine to make a solution…

Answer 20

m₁ + m₂

Question 21

…gained 60 milligrams…

Question 22

Changing words into algebraic expressions is necessary to solve word problems.

What are some of the words you can name that mean the subtraction operation?

Answer 22

Here are some words that translate into subtraction:

• remove
• deduct
• depriciate
• shorten
• drop
• lose
• lower
• left
• difference
• decreased by
• fewer
• less than
• smaller than
• take away

Question 23

Changing words into algebraic expressions is necessary to solve word problems.

What are some of the words you can name that mean multiplication operation?

Answer 23

Here are some words that translate into multiplication:

• double
• twice
• triple
• quadruple
• times
• squared
• cubed
• factor
• product
• multiple of

Question 24

Changing words into algebraic expressions is necessary to solve word problems.

What are some of the words you can name that mean division operation?

Answer 24

Here are some words that translate into division:

• ratio
• average
• split
• half
• per
• quotient
• third
• fourth
• part of

Question 25

Define:

The distance between two real numbers on the number line.

d (a,b)

Answer 25

The distance between two real numbers a and b on the number line is defined as the absolute value of b - a.

• d (a,b) =* Ib - aI
• Example:*

d (5, -3) = I-3 - 5I = 8

Question 26

a - b

Answer 26

When you add two numbers with different signs, subtract the absolute values (the smaller from the larger) and keep the sign of the number with the greater absolute value.

• Example:*
• 5 + 3 = -2

|-5| - |3| = 5 - 3 = 2

Since -5 has the greater absolute value than 3, the result is negative.

Question 27

What number is the negative of a negative number?

-(-a) = ?

Answer 27

The negative of a negative number is the opposite positive number.

• (-a) = + a

Question 28

Find solutions to the equation below.

I2x - 5I = 7

Answer 28

x = 6; x = -1

The absolute value of 2x - 5 equals 7. It means that:

2x - 5 = 7 ⇒ 2x = 12 ⇒ x = 6

2x - 5 = -7 ⇒ 2x = -2 ⇒ x = -1

Question 29

Solve:

• (- 4 - 7) + (- 2) = ?

Answer 29

• (- 4 - 7) + (- 2) = - (-11) + (-2) = 11 - 2 = 9.

Question 30

• x, y* and z are distinct integers.
• x < y < z*
• y < 0*
• z = xy*

Is z positive or negative?

Answer 30

Z is positive.

Since x < y and y < 0, both x and y must be negative integers. The product of two negative integers is a positive integer.

Question 31

If there are two positive numbers and three negative numbers in a multiplication sequence, what sign is the product of the five numbers?

Answer 31

The product is negative.

positive x positive = positive

negative x negative = positive

positive x negative = negative

Example:

2 x (-1) x 2 x (-3) x (-1) = -12

Question 32

What is the solution set of the inequality below?

I2x - 5I > 7

Answer 32

• 1 > x > 6

The absolute value of 2x - 5 is greater than 7. It means that:

2x - 5 > 7 ⇒ 2x > 12 ⇒ x > 6

2x - 5 > -7 ⇒ 2x > -2 ⇒ x < -1

*** Remember, dividing by a negative number changes the direction of the inequality sign.

Question 33

Can you determine the sign of the result of the expression below?

(-5) x 6 x (-4) x 0 x 8 = ?

Answer 33

No, you cannot. The result is zero and it doesn’t have a sign. Zero is neither negative nor positive.

Question 34

Find the quotient and determine its sign.

• ( - 132) / 6 = ?

Answer 34

22

The quotient is positive.