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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).

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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).