ACT Algebra Absolute Value

SAT Question of the Day

The SAT question of the day is an Identifying Sentence Errors Question that has already been addressed on this blog: click here to see an explanation.

ACT Math Question of the Day

What are the values of a and b, if any, where a|b – 2| < 0 ?

There are three things you must understand to solve this problem: number properties, absolute value, and inequalities.

Start by considering the equation as a whole. You know that the side that contains the variables must be less than zero. What kind of numbers are less than zero? Negative numbers! Now you know that a|b – 2| must be a negative number.

Break down a|b – 2| into its essential parts. It is really just the variable a multiplied by |b – 2|. What kinds of numbers do you have to multiply together in order to get a negative number? You must have a positive number times a negative number. You know that anything inside the absolute value bars will be positive. That is your positive number. Now you know that variable a must be negative! In other words a < 0!

Now that you know the restrictions on variable a, look at b. You know that you need a negative number times a positive number and it must be less than 0. Think about the properties of 0. Zero is neither negative nor positive. You must make sure that the second part of your equation (|b – 2|) does not equal 0. When does |b – 2| = 0? When b = 2. Therefore, for your equation to work, b ≠ 2.

A.a < 0 and b ≠ 2
B.a < 0 and b = 2
C.a ≠ 0 and b > 2
D.a > 0 and b < 2
E. There are no such values of a and b.

The correct answer is (A).

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Number Properties

Arithmetic: Number Properties

Read the following SAT test question and then select the correct answer.

Every time you work a math problem, read the problem carefully.  Identify the bottom line and think about the most efficient method to solve for the bottom line.  Choose a method to solve the problem and attack the problem without hesitation.  When you think you have the answer, loop back to make sure that the answer addresses the bottom line because questions often require multiple steps to get to the answer.

When the positive integer n is divided by 5, the remainder is 0. What is the remainder when 3n is divided by 5?

Make a note that your bottom line is the remainder of 3n.  Then think carefully about the first portion of information that you are given.  Some number, n, is divided by 5 and there is no remainder.  That means that n must be a multiple of 5.  If you do not immediately see this, think about a concrete number that will not result in a remainder:

 or    or  
All of these values for n result in a whole number with no remainder.

If n is a multiple of 5, what will 3n be?  It will still be a multiple of 5!  It will still result in a remainder of 0.  If you cannot see this, look back at the examples above using 5 and 15.  If 5 is your n value, 3 times 5 is 15, and when you divide 15 by 5 the answer is 3 with a remainder of 0.  Now that you have found your bottom line, look down at your answer choices.  

(A) 0
(B) 1
(C) 2
(D) 3
(E) 4

The correct answer is (A).

On, 70% of the responses were correct.

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Number Properties

Read the following SAT test question and then select the correct answer. 

Read the question carefully so that you are sure you understand what you are being asked.  You must identify the bottom line that you will solve for; note it at the top of your scratch work.  Assess your options to find the most efficient way to solve the problem, and then attack the problem.  Be sure to write out your scratch work clearly so that you do not make careless mistakes.  Your last step is to loop back to make sure that your answer matches the bottom line.  You cannot get a question right if you solved for an answer that you were not asked to find.

For how many positive two-digit integers is the ones digit greater than twice the tens digit?

You must find out how many numbers fit the given requirements.  There is no formula to find numbers in which one digit is more than twice the other, so you must think about this question logically.  Your only option is to methodically check positive integers to see which numbers will work.

You know that you need a positive two digit integer, so your first digit, at the very least, must be a 1.  Your second digit must be more than twice the first.  The number 1 multiplied by 2 is 2, so the number 12 will not fit the requirement of having a ones digit greater than twice the tens digit.   However, any number that begins with a 1 and has a second digit larger than 2 will work.  List all the numbers that begin with 1 and fit the requirements of this problem:

13, 14, 15, 16, 17, 18, 19

After 19, you must start each number with a 2, so find out what the second digit must be.  Again, it must be bigger than twice the first digit, so it must be larger than 4.  24 will not work, so start with 25 and list all of the numbers that fit the requirements of this problem:

 25, 26, 27, 28, 29

Follow this procedure for numbers beginning with 3. 3 times 2 is 6, so only numbers larger than 36 will work.

37, 38, 39

Move on to numbers that start with a 4.  4 times 2 is 8, so the second digit must be greater than 8.  This time there is only one number that fits the requirements:


Now you have reached numbers beginning with the digit of 5.  5 times 2 is 10.  You cannot have a value as your second digit that is more 10, so any number larger than 49 will not work.

Count up all of the numbers that you have found that fit the requirements of this problem.  That number will satisfy your bottom line.

(A) 16
(B) 20
(C) 28
(D) 32
(E) 36

The correct answer is (A).

On, 40% of the responses were correct.

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Number Properties

Read the following SAT test question and then select the correct answer. 

In the SAT math section, you must read every problem carefully and identify the bottom line.  Assess your options before solving the problem so that you are able to choose the most efficient method of solving the problem.  Then attack the problem to find your answer, and loop back to make sure that your answer addresses the bottom line.

The sum, product, and average (arithmetic mean) of three integers are equal. If two of the integers are 0 and -5, the third integer is…

This problem asks you to find one unknown integer.  You could try to write equations or plug in the answer choices to solve this problem.  However, these methods will take you longer than thinking logically about the properties of the numbers involved.

You know that the sum, product, and average of three integers must be equal.  One of the numbers that you are given is a zero.  Zero multiplied by any other number will always be zero, so the product must be zero.  That means that the sum and the average of these three numbers must also be zero. What do you have to add to 0 and -5 in order to get zero? The only possible answer is a positive 5.  Additionally, if you add 0, -5, and 5 together and then average them, your sum already equals zero, so zero divided by 3 will still be zero.  The product, sum, and average are all 0 when the missing third integer is 5.

0 + -5 + x = 0
x = 5

(0 + -5 + x) / 3 = 0
x = 5

Before looking at the answer choices, check to make sure that your answer fits the bottom line that you were asked to find.

(A) -5
(B) 0
(C) 2
(D) 5
(E) 10

The correct answer is (D).

On, 72% of responses were correct.

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Link of the Day

Pythagoras, best known to high school students for his Pythagorean Theorem, actually discovered much more than that one formula. Even if you are not mathematically inclined, the beginning of this paper has some interesting notes on how the Pythagoreans--the followers of Pythagoras--lived. 

3/28 Integers

Read the following SAT question and then select your answer.

Always attempt to solve the problem before looking at the answer choices. Read carefully, then identify the bottom line--what the question is actually asking--and mark it at the top of your scratch work. Assess you options by asking "What could I do?" to open your toolbox, then "What should I do?" to select the best way to solve the problem. Attack the problem fearlessly, then loop back to the bottom line to check whether what you found is the correct answer. 

If p is an odd integer, which of the following is an even integer?

At the top of your scratch work, write even = ?

Next, ask "What could I do?" You could think through each answer choice abstractly, determining that if p is odd then... but that is difficult and gets confusing quickly. You could pick a number for p, then use that number to find a value for each answer choice. The smallest odd number is the best for this. Pick one. Since this question includes the phrase "which of the following," the answer is very likely to be D or E. Start at the bottom and work your way up.

If p = 1, then . 0 is neither positive nor negative, but neutral; however, it is still even. This distinction confuses some students, so make sure you know it. Now loop back to the bottom line. , so it is even, so it is the answer. On the SAT, you could continue on from this point or check the other answers.

 is odd.

 is odd.

is odd.

 is odd.

The answer is E.

On, 62% of responses were correct.

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