# Inequalities

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

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 $\frac{2}{5}$.

A) 0

B) $\frac{1}{5}$

C) $\frac{2}{5}$

D) $\frac{3}{5}$

E) $\frac{4}{5}$

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 sat.collegeboard.org, 52% of responses were correct.

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