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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Algebra: Inequalities

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

Always read the question carefully, identifying the bottom line.  Assess your options for reaching the bottom line and use the most efficient method to attack the problem.  When you have an answer, loop back to verify that your answer matches the bottom line.


On the line above, if AB < BC < CD < DE, which of the following must be true?

Bottom Line: wotf must be true = ? (which of the following)

Assess your Options:  For a “wotf” question, you will have to look at the answer choices.  Most students will start with “A,” so Knowsys recommends that you start with “E.”  You may also find that this is a good problem to use the strategy of plugging in numbers.

Attack the problem:  Take a look at your answer choices:
(A) AC < CD
(B) AC < CE
(C) AD < CE
(D) AD < DE
(E) BD < DE

(E) BD < DE  Look up at the figure.  On the figure, does BD look smaller than DE?  No!  It looks slightly larger.  You know that the figure is not drawn to scale, but the figure does give you one possible depiction of the rule.  Use the figure!  If it is possible for BD to be bigger than DE, then this answer is incorrect because you are looking for something that must be true.  Eliminate this choice.

(D) AD < DE  Look up at the figure.  The figure shows you that it is possible for AD to be larger than DE.  Eliminate this choice.

(C) AD < CE  These lengths are very similar on the line.  Break each length down into the parts that compose it so that you can make a precise comparison.  For example, AD contains AB + BC + CD.  CE contains CD + CE.  You now have: AB + BC + CD < CD + DE.  When you have the same thing on both sides of an equation, it cancels.  Eliminate the CD.  You now have AB + BC < DE.
You cannot come to a conclusion about these lengths.  If you want to prove this, try plugging in numbers.  Suppose AB starts at 10 and each portion along this line gets larger by 1.  AB = 10, BC = 11, CD = 12, DE = 13.  Is 10 + 11 < 13?  No.  Eliminate this choice.
(B) AC < CE  This one looks like it could be true, based on the figure.  See if you can prove it.  Break it down into its parts just as you broke down the last answer choice.  AC contains AB + BC.  CE contains CD + DE.  At first it seems as if you cannot compare these either because all of the numbers are different.  Try plugging in the same values as you used before: AB = 10, BC = 11, CD = 12, DE = 13.  Is 10 + 11 < 12 + 13?  Yes!  Will this work for all numbers?  Yes!  You are adding a small number plus a medium number and comparing it to a big number plus an even bigger number.  The former will always be smaller than the latter.  Once you know this, you do not even need to check (A).

(A) AC < CD You can tell from the figure that this does not have to be true.

Loop back:  You solved for what must be true, so you should select the answer you found.

The correct answer is (B).

On, 68% of the responses were correct.

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ACT Question of the Day:

If you have gone 4.8 miles in 24 minutes, what was your average speed, in miles per hour?

Your bottom line here is in miles per hour.  That would be miles over hours.  Your distance (miles) is in the correct unit, but your time (minutes) is not.  You know that there are 60 minutes in an hour.   Find the fraction of an hour that was spent traveling. Take your minutes and put them over the total minutes in an hour:

Now you know that you went 4.8 miles in .4 hours.  How many miles per hour was that?  Divide 4.8 by .4 and you will see that the answer is 12. 

Note: You can do this in your head if you realize that this is the same thing as dividing 48 by 4.  This whole problem can be done in seconds if you know your times table all the way up to 12. 

Look down at your answer choices.

(A)  5.0
(B) 10.0
(C) 12.0
(D) 19.2
(E) 50.0

The correct answer is (C).

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Link of the Day

One of the best ways to improve your SAT essay score is to begin looking for links between ideas.  Even though you don’t know what the SAT essay will be about, it will be broad enough to relate to news articles and books that you are reading now.  Practice looking for themes in this article about Iranian hostages who were just recently released.  You should be able to list the themes: family, politics, government, individuals, beliefs, and many more.  As soon as you see an essay question relating to any of these themes, you can use the facts from this article as an excellent example in support of your thesis.  If you choose to use this as one of your 5 prepared current events, make a list of all the facts you can use.

Arithmetic: Percents

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

Always read math questions carefully so that you can absorb all the information and avoid mistakes.  Identify the bottom line, what the question is asking you to find, and assess your options for reaching that bottom line.  Choose the most efficient method to attack the problem.  When you have an answer, loop back to verify that the answer matches the bottom line and you have finished all the steps in the problem.

If p percent of 75 is greater than 75, which of the following must be true?

Bottom Line: p = ?

Assess your Options:  It is often tempting to look down at the answer choices before you need them, but they could mislead you since most of them are wrong!.  You could take numbers that fit each answer choice and see if they give you a number greater than 75.  However, by applying what you know about percents, you can solve the problem much faster than you can by trying out 5 different numbers.

Attack the Problem:  There are a number of ways to think about percentages: as percents, decimals, numbers out of a hundred, parts of wholes….  The list continues.  Here is one of the fastest ways to think about the problem:

If you have one hundred percent of something, you have all of it.  So 100% of 75 is going to be 75.  If you want a result that is greater than 75, you are going to need more than 100% of 75.  Therefore, p must be bigger than 100.

Or, if you normally think about percents in terms of decimals, you know that 50% of something is .5.  In order to get a decimal from a percent, you had to move the decimal twice to the left.  So with 100%: 75 × 1.00 = 75.  Try writing an inequality to find the decimal that you would need in order to get a number bigger than 75: 75p > 75.  The p represents the unknown percent of 75 (remember, "of" means multiplication in math).  If you solve the inequality, you get p > 1.  Then you have to move the decimal back in order to get a percent: p > 100.  Your percent must be bigger than 100%.  This method takes much longer than the first one, but it proves that the first method is correct.  The testers realize that students are not used to working with percentages greater than 100, so it is a good idea to review how these work before the test!

Loop back:  You know what p must be greater than, so look down at your answer choices.

(A) p > 100
(B) p < 75
(C) p = 75
(D) p < 25
(E) p = 25

The correct answer is (A).

On, 71% of the responses were correct.

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Data Analysis: Probability

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

Read each question carefully to avoid making any mistakes. Identify the bottom line (what the question is asking) and assess your options for reaching it by asking yourself “What could I do?” and “What should I do?” Choose the most efficient method to attack the problem and find an answer. Last, loop back to make sure that your answer addresses the bottom line.

If a number is chosen at random from the set {-10, -5, 0, 5, 10}, what is the probability that it is a member of the solution set of both 3x – 2 <10 and x + 2?

Bottom Line: Prb = ?

Assess Your Options: You cannot solve for a probability until you know whether each number in the set meets the requirements that you are given. You could plug numbers from the set into each inequality and see if they work, but it is much faster to simplify the inequalities before you begin working with them.

Attack the Problem: Simplify the inequalities by solving both for x.

3x – 2 < 10
3x < 12
x < 4

x + 2 > -8
x > -10

You now know that x must be less than 4, but greater than -10. The question asked you to find a number that fits both of these solution sets. Look at the original set that you were given. The only two answers that are between -10 and 4 are -5 and 0 (-10 does not work because it cannot be equal to negative -10; it has to be greater than -10). You found 2 numbers out of 5 that you were given that work. To write this as a probability, you must set the number of relevant outcomes over the number of total possible outcomes. Your answer is 2 over 5.

Loop Back: You found a probability matching the restrictions you were given. Look down at your answer choices.

(A) 0

(B) 1 over 5

(C)2 over 5

(D)3 over 5

(E)4 over 5

The correct answer is (C).

On, 50% of the responses were correct.

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Link of the Day

This page lists some of the great mathematicians of the ages, including Newton, Archimedes, Euclid, and others. Using any of them in an essay will help you stand out and earn a higher score. 

4/9 Inequalities

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

Be sure to read carefully--reading carelessly will cost you points. Mark the bottom line and assess your options, then choose the fastest route to the answer. Remember: The long way is the wrong way! After you find an answer, check it against your bottom line before looking among the answer choices for the number you found.

If a number is chosen at random from the set {-10, -5, 0, 5, 10}, what is the probability that it is a member of the solution set of both 3x - 2 < 10 and x + 2 > -8?

First, look for your bottom line. It is in two parts here, so one thing you could do is try to combine them to simplify the problem. You could also find the numbers that satisfy each half of the bottom line and then combine them. The long way is the wrong way, so you will need to combine the inequalities. First, isolate x.

p (both inequalities) = ?

3x - 2 < 10                                                               x + 2 > -8

3x < 12                                                                    x > -10

x < 4

Next, you can combine the two inequalities into one compound inequality.

-10 < x < 4

Now look for the values of the given set that satisfy the inequality. There are only two: -5 and 0. Since the set has 5 terms, the probability that any random term is one of the two you found is .

A) 0





The answer is C.

Alternate Method: If rearranging the inequalities doesn't occur to you, don't sweat it. If it takes longer to remember one tool than it does to use a different one, then that isn't the right tool. Instead, after isolating x in each inequality, you can check to see which numbers fit each of the two inequalities you have.

x < 4                  -10, -5, 0

x < -10               -5, 0, 5, 10

Only two numbers fit both inequalities: 0 and -5. Again, these are two numbers out of five, so your probability and your answer are the same.

On, 52% of responses were correct.

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Link of the Day: The History Channel's website is full of fascinating articles and videos. Watch their current series, "Full Metal Jousting," and learn some math facts about St. Patrick's Day.

3/22 Functions

Read the following SAT question and then select your answer. 

Remember to follow the Knowsys Method: Read carefully, identify the bottom line, and assess your options. After you ask, "What could I do?" and "What should I do?" attack the problem, then loop back to check whether you answered the question correctly. Finally, take a look at the answer choices and select the correct one.

math image

The graph of y=f(x) is shown above. If , and if (t,v) is on the graph of f, which of the following must be true?

A glance at the answer choices shows that they all have v, so v=? will do for an imperfect bottom line. Now look at the question again. If t is somewhere from 0 to 5, what does that say about v? Note that t is the x value, and v is the y value, so you need to focus on the part of the graph from x=0 to x=5. In that range, the smallest y value is 5 and the greatest is 10. Now look at the answer choices.

The answer is D.

On, 56% of responses were correct.

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