ACT Math

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).


For the ACT Question of the Day, visit http://www.act.org/qotd/.

To get help preparing for the SAT, PSAT, or ACT Exam, visit www.myknowsys.com!

Absolute Value

Link of the Day

Happy St. Patrick’s Day!  If you like celebrating holidays, consider tracing the history of a holiday celebration as one of your historical events, or even tracing the history of a holiday up to the present for a current event.  There are many misconceptions about the origins of ideas and traditions, and the way that certain practices came about is fascinating.  Think about how this article about the history of St. Patrick’s day could be used to answer the following SAT essay questions:

(1)  Should people change their decisions when circumstances change, or is it best for them to stick with their original decisions?

(2)  Do you think that ease does not challenge us and that we need adversity to help us discover who we are?

(3)  What motivates people to change?

Algebra: Absolute Value

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

If , which of the following could be true?

Bottom Line: which of the following COULD be true?

Assess your Options:  You could come up with answers to solve this problem, but that will be a waste of time if they are not included in your answer choices.  Instead, look at the answer choices and methodically eliminate incorrect answers.

Attack the Problem:  Look at your answer choices:

(A) a = 0

(B) b = 0

(C) a = b

(D) a = -b

(E) a = 1

For a “which of the following” question, Knowsys recommends that you begin with answer choice (E).

Hint:  Instead of thinking of the bars in your equation as absolute value, think of them as simply showing where a positive number will be.  If you do this, you will not have to plug in actual numbers and you can check each answer choice using logic. (This works because an absolute value simply tells you how far a number is from zero.  The double bars only affect negative numbers, making them positive.)

(E) a = 1    Plug a = 1 into your original equation.   Is there any way to start out with the number 1 and subtract a positive number to get the answer 5?  There is not.  Eliminate this answer.

(D) a = -b     This answer choice has a negative sign, but remember that any negative sign will go within the bars and come out a positive number.  So if you plug in b where the variable a is in this equation, you still end up with b – b = 5.  Is that possible?  No!  Anything minus itself will be zero.  Eliminate this answer.

(C) a = b     This answer is essentially the same as the last one!  If you plug in b where you have an a, you wind up with b – b = 5.  Again, anything minus itself will be zero.  Eliminate this answer.

(B) b = 0     Plug in 0 for the b in your equation.  You now have a positive number minus 0 equals 5.  Is that possible?  Yes!  5 – 0 = 5.  You are finished.  You don’t have to know that a can be either negative 5 or 5, and you don’t have to check the last answer choice.  Let’s check it just for practice.

(A) a = 0     Is there any way to start with 0 and subtract a positive number to get 5?  No!  Eliminate this choice.

The correct answer is (B).

On sat.collegeboard.org, 63% of the responses were correct.

For more help with SAT math, visit www.myknowsys.com!

Picking Numbers

Mathematics: Standard Multiple Choice

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

Multiple-choices directions are really simple, aren't they? The questions on the test are longer, but they don't say much more than this. Remember that you are allowed to use your calculator, but that it can get you in trouble. I don't actually own one, and I use the one on my computer as little as possible. 

The weights of 10 bags of apples range from 2.75 pounds to 3.15 pounds. If w is the weight, in pounds, of one of these bags, which of the following must be true?

After reading carefully, the next step is to identify the bottom line. Here, the question asks "Which of the following must be true?" and that's extremely hard to put in mathematical shorthand, so the best choice here is to write weight=? at the top of your scratch work.

Next, assess your options. You could translate this problem into an inequality, but that could be complicated because the answers are all absolute value. Instead, pick numbers for w and see which inequality below works out. To make sure that the answer choice will always be true, try three different numbers. When picking numbers, always choose numbers that are easy to work with and that are distinct from one another. In this case, picking 3 and then picking 3 again won't tell you anything about the problem.   3, 2.8, and 3.1 should work nicely. I picked 3 first because it is a whole number, and those are always easier to deal with. 2.8 and 3.1 are near the ends of the given range, so they will make sure you get the right answer.