Translating Words to Absolute Values - SAT Math

A helpful strategy on problems that ask you to convert a situation to an absolute value is to pick numbers that would work and then test the answer choices. Here if you were using numbers and not variables and absolute values this isn't that hard a problem: in order to win, you need to guess between 227 and 237 so that your number is within 5 of the total of 232. So you want an absolute value that will work for both 227 and 237. if you then test the answer choices you should see that the correct answer, , gives you the right boundaries for winning:

when gives you and when gives you . This shows you that this absolute value works: if you're between -5 away and 5 away from the total number, you win.

Translating word problems to absolute values can be tricky, but in a multiple choice format you have a great opportunity to use process of elimination. Here the wording should lead you to a pretty good understanding of the actual numbers involved in the scenario: Josh's guess was $6 off of the correct value of $112, so he must have guessed either $106 or $118. So you can plug in those values for and see which answer choices match with both of those values. Among the answer choices:

works. 118-112 = 6 and 106-112 = -6, for which the absolute value is 6.

does not work. 118-6 does equal 112, but 106-6 = 100, for which the absolute value is not 112.

does not work. 112-6 does equal 106, but it does not equal 118 (or -118) so this equation doesn't account for both possibilities.

does not work. 6-106 = -100, so the absolute value does not match 112.

When you're converting words to absolute values in a multiple choice problem, it is often easiest to use process of elimination by testing the answer choices. Here you should see that any board between 35.95 and 40.05 inches, inclusive, would be kept, but any board shorter than 35.95 or longer than 40.05 inches would be discarded.

If you then use those values in testing the answers, you're looking for an answer choice that satisfies both cases.

does not work. A board of 35.94 should be rejected, but here it would not be, as 35.94 + 0.05 = 35.99, and that would not fit this inequality.

is the correct answer. A board of 35.94 inches would mean that 35.94-36 = -0.06, and therefore the absolute value would be greater than 0.05. Or a board of 36.06 would give an absolute value of 0.06 as well, again satisfying the inequality.

should clearly look wrong, as any reasonable-length board that should be kept would not fit this inequality (a board of exactly 36 inches would mean that the absolute value is 72, and that's nowhere near "less than 0.05").

is also incorrect. A board of 36 inches should not be rejected, but here 36.05 > 36 would suggest that it would be.

A helpful way to think about absolute values involves the number line. The general definition of absolute value is "distance from zero" (which is why you'd take, for example, -5 and say that its absolute value is just 5, because -5 is "5 away from zero"). Here you can think of "distance from 32" as your goal: if the value is within one increment of 32, then you'll keep it, but if it's more than one increment away from 32 the bag will be rejected. That means that a rejected bag will have a difference of more than 1 from 32.

That's what the correct answer essentially says: means that "the positive difference between and 32 is more than 1." Breaking this apart, is "the difference between and 32," and the absolute value means that we don't care whether it's a positive or negative difference - we'll just take whatever difference we get and use the positive version of it, since being too light is as big an issue as being too heavy. And then is there to show that if the difference is within 1 it doesn't fit this definition of "bag to be discarded," but as soon as that difference is greater than 1 that bag will be discarded.

Often times in word problems involving inequalities it's fairly straightforward to work with the numbers, and then more challenging to work with the algebra. Here you should be able to determine that any number of almonds between 27 and 33, inclusive, would be satisfactory for the trail mix company. So how can you use those numbers? Plug those in as values to each answer choice and make sure that both values work; for any answer choice in which those numbers do not work, you can eliminate it. And if the numbers do work, try a clearly not-allowed number to see if it fits the inequality or not; if it fits, you know that the inequality allows for unallowable numbers, so you can eliminate that option, too.

does work for both 27 and 33, but notice how they're not even close to the borderline. For example 33 + 30 is 63 which is way, way more than 3. If you were to try a not-allowed number of almonds, then, like 25, you should see that this inequality doesn't work. 25 + 30 is far greater than 3, which would seem to suggest that 25 is an allowable number when we know that it is not.

similarly fails the test for numbers we know are not allowed. If you try 25 again, it would work in this inequality but it is not an allowable answer.

works. Both 27 and 33 will give an absolute value of 3, and you know that that is the limit. If you try a not-allowed number like 25, you'll see that it does not fit so this inequality does a good job of only keeping allowable numbers.

Does not work for 27, which you know to be an allowable number. 27-3 is 24, which is not greater than 30, so this inequality omits an allowable outcome and is therefore incorrect.

Often on word problems that translate to absolute values, it is fairly straightforward to find the exact numbers, but trickier to set up the algebra. You can often then find the numbers that will work and plug those in to the answer choices to use process of elimination. Here if Jesse's birthday is November 17, you should see that the only two dates 7 days away are November 10 and 24. So you know that must be either 10 or 24. If you plug in both of those values to each answer choice, you can eliminate any choice for which either value does not work.

works for both values. and

does not work for November 10, as .

does not work for November 24, as .

does not work for either value.

Often times on problems that ask you to translate words to absolute values, it is fairly straightforward to find values that are and are not allowed by the situation, and that allows you to use process of elimination by plugging those numbers in. Here the numbers 62 and 70 should work, and the numbers 60 and 72 should not work. You can then test the answer choices:

does work for 62 and 70, but note that it also works for 72 while it should not. This answer is incorrect.

works for both 62 and 70, and does not work for 60 and 72. This is the correct answer.

works for 70 but not for 62, so this answer is incorrect.

works for both 62 and 70, but it also works for 72 and even higher numbers, so it is incorrect.

When you think about the situation that you're presented here, you know that Hannah's guess was exactly four places away from the exact number of 11,454 (so she guessed either 11,450 or 11,458). What this means in a mathematical sense is that the difference between her guess, , and 11,454 was 4. Now, you could almost say that "the difference" means subtraction so just . But we don't know whether her number was greater or less than 11,454. Here's where the absolute value comes in; if you were to say that (the correct answer), you'd know that the positive difference of her number and the real number was 4. The positive difference allows you to not have to guess whether it was or that exactly equaled 4.

Of course, on these questions where it's fairly straightforward to find the numbers but markedly harder to set up the algebra, you can always find the numbers that work (here 11,450 and 11,458) and then plug those into the answer choices to use process of elimination. Here only works for both cases, so it is the correct answer.

This problem shows the importance of using absolute values in certain word problems. Here you know that Jacob and Philip were born 4 years apart, but do you know who is older? If you look at the equations and , either could be true depending on which son is older. If Jacob is older, then , but if Philip is older then .

properly accounts for that uncertainty; it says that the difference between their ages is 4, but does not require knowledge of who is older.

Often on problems that ask you to translate words to absolute values, it's fairly straightforward to choose numbers that will work in the word problem, but more challenging to create the algebra. In those cases, you can use process of elimination by determining values that do and do not work for the situation, and then plugging them into the answer choices. Here, 55 and 65 are both allowable values of , but values like 50 and 70 are not. If you plug those into the answer choices:

works. 55 and 65 each work, and are the limits (for each, the absolute value equals 5). 50 and 70 do not work.

does not work. You know that 65 is an allowable number but when you plug that in here the inequality does not hold.

does not work. You know that 55 is an allowable number but when you plug that in here the inequality does not work.

does not work, as no allowable numbers when added to 60 are less than or equal to 5.

Often on problems that ask you to translate words to absolute values, it's fairly straightforward to choose numbers that will work in the word problem, but more challenging to create the algebra. In those cases, you can use process of elimination by determining values that do and do not work for the situation, and then plugging them into the answer choices. Here, 55 and 65 are both allowable values of , but values like 50 and 70 are not. If you plug those into the answer choices:

works. 55 and 65 each work, and are the limits (for each, the absolute value equals 5). 50 and 70 do not work.

does not work. You know that 65 is an allowable number but when you plug that in here the inequality does not hold.

does not work. You know that 55 is an allowable number but when you plug that in here the inequality does not work.

does not work, as no allowable numbers when added to 60 are less than or equal to 5.

A helpful strategy on problems that ask you to convert a situation to an absolute value is to pick numbers that would work and then test the answer choices. Here if you were using numbers and not variables and absolute values this isn't that hard a problem: in order to win, you need to guess between 227 and 237 so that your number is within 5 of the total of 232. So you want an absolute value that will work for both 227 and 237. if you then test the answer choices you should see that the correct answer, , gives you the right boundaries for winning:

when gives you and when gives you . This shows you that this absolute value works: if you're between -5 away and 5 away from the total number, you win.

Translating word problems to absolute values can be tricky, but in a multiple choice format you have a great opportunity to use process of elimination. Here the wording should lead you to a pretty good understanding of the actual numbers involved in the scenario: Josh's guess was $6 off of the correct value of $112, so he must have guessed either $106 or $118. So you can plug in those values for and see which answer choices match with both of those values. Among the answer choices:

works. 118-112 = 6 and 106-112 = -6, for which the absolute value is 6.

does not work. 118-6 does equal 112, but 106-6 = 100, for which the absolute value is not 112.

does not work. 112-6 does equal 106, but it does not equal 118 (or -118) so this equation doesn't account for both possibilities.

does not work. 6-106 = -100, so the absolute value does not match 112.

When you're converting words to absolute values in a multiple choice problem, it is often easiest to use process of elimination by testing the answer choices. Here you should see that any board between 35.95 and 40.05 inches, inclusive, would be kept, but any board shorter than 35.95 or longer than 40.05 inches would be discarded.

If you then use those values in testing the answers, you're looking for an answer choice that satisfies both cases.

does not work. A board of 35.94 should be rejected, but here it would not be, as 35.94 + 0.05 = 35.99, and that would not fit this inequality.

is the correct answer. A board of 35.94 inches would mean that 35.94-36 = -0.06, and therefore the absolute value would be greater than 0.05. Or a board of 36.06 would give an absolute value of 0.06 as well, again satisfying the inequality.

should clearly look wrong, as any reasonable-length board that should be kept would not fit this inequality (a board of exactly 36 inches would mean that the absolute value is 72, and that's nowhere near "less than 0.05").

is also incorrect. A board of 36 inches should not be rejected, but here 36.05 > 36 would suggest that it would be.

A helpful way to think about absolute values involves the number line. The general definition of absolute value is "distance from zero" (which is why you'd take, for example, -5 and say that its absolute value is just 5, because -5 is "5 away from zero"). Here you can think of "distance from 32" as your goal: if the value is within one increment of 32, then you'll keep it, but if it's more than one increment away from 32 the bag will be rejected. That means that a rejected bag will have a difference of more than 1 from 32.

That's what the correct answer essentially says: means that "the positive difference between and 32 is more than 1." Breaking this apart, is "the difference between and 32," and the absolute value means that we don't care whether it's a positive or negative difference - we'll just take whatever difference we get and use the positive version of it, since being too light is as big an issue as being too heavy. And then is there to show that if the difference is within 1 it doesn't fit this definition of "bag to be discarded," but as soon as that difference is greater than 1 that bag will be discarded.

Often times in word problems involving inequalities it's fairly straightforward to work with the numbers, and then more challenging to work with the algebra. Here you should be able to determine that any number of almonds between 27 and 33, inclusive, would be satisfactory for the trail mix company. So how can you use those numbers? Plug those in as values to each answer choice and make sure that both values work; for any answer choice in which those numbers do not work, you can eliminate it. And if the numbers do work, try a clearly not-allowed number to see if it fits the inequality or not; if it fits, you know that the inequality allows for unallowable numbers, so you can eliminate that option, too.

does work for both 27 and 33, but notice how they're not even close to the borderline. For example 33 + 30 is 63 which is way, way more than 3. If you were to try a not-allowed number of almonds, then, like 25, you should see that this inequality doesn't work. 25 + 30 is far greater than 3, which would seem to suggest that 25 is an allowable number when we know that it is not.

similarly fails the test for numbers we know are not allowed. If you try 25 again, it would work in this inequality but it is not an allowable answer.

works. Both 27 and 33 will give an absolute value of 3, and you know that that is the limit. If you try a not-allowed number like 25, you'll see that it does not fit so this inequality does a good job of only keeping allowable numbers.

Does not work for 27, which you know to be an allowable number. 27-3 is 24, which is not greater than 30, so this inequality omits an allowable outcome and is therefore incorrect.

Often on word problems that translate to absolute values, it is fairly straightforward to find the exact numbers, but trickier to set up the algebra. You can often then find the numbers that will work and plug those in to the answer choices to use process of elimination. Here if Jesse's birthday is November 17, you should see that the only two dates 7 days away are November 10 and 24. So you know that must be either 10 or 24. If you plug in both of those values to each answer choice, you can eliminate any choice for which either value does not work.

works for both values. and

does not work for November 10, as .

does not work for November 24, as .

does not work for either value.

Often times on problems that ask you to translate words to absolute values, it is fairly straightforward to find values that are and are not allowed by the situation, and that allows you to use process of elimination by plugging those numbers in. Here the numbers 62 and 70 should work, and the numbers 60 and 72 should not work. You can then test the answer choices:

does work for 62 and 70, but note that it also works for 72 while it should not. This answer is incorrect.

works for both 62 and 70, and does not work for 60 and 72. This is the correct answer.

works for 70 but not for 62, so this answer is incorrect.

works for both 62 and 70, but it also works for 72 and even higher numbers, so it is incorrect.

When you think about the situation that you're presented here, you know that Hannah's guess was exactly four places away from the exact number of 11,454 (so she guessed either 11,450 or 11,458). What this means in a mathematical sense is that the difference between her guess, , and 11,454 was 4. Now, you could almost say that "the difference" means subtraction so just . But we don't know whether her number was greater or less than 11,454. Here's where the absolute value comes in; if you were to say that (the correct answer), you'd know that the positive difference of her number and the real number was 4. The positive difference allows you to not have to guess whether it was or that exactly equaled 4.

Of course, on these questions where it's fairly straightforward to find the numbers but markedly harder to set up the algebra, you can always find the numbers that work (here 11,450 and 11,458) and then plug those into the answer choices to use process of elimination. Here only works for both cases, so it is the correct answer.

This problem shows the importance of using absolute values in certain word problems. Here you know that Jacob and Philip were born 4 years apart, but do you know who is older? If you look at the equations and , either could be true depending on which son is older. If Jacob is older, then , but if Philip is older then .

properly accounts for that uncertainty; it says that the difference between their ages is 4, but does not require knowledge of who is older.