Inequalities and Ranges
When the equal sign in an equation is replaced by a less than (<) or greater than (>) sign, you have an inequality. You can still manipulate both sides of the inequality, and you can still distribute and factor to your heart’s content. In fact, there is only one thing you have to remember when dealing with inequalities: If you multiply or divide both sides by a negative number, switch the direction of the inequality sign.
Now here’s a pop question to test your SAT savvy. Because this is the one weird thing about inequalities, how often do you think it pops up on the SAT? Your answer choices are Never, Sometimes, and All the freakin’ time. No hurry. We’ll wait.
If you answered “all the freakin’ time,” bully for you. You’re starting to get the hang of the test. You see, almost every SAT item has a catch to it. If you know the catch, the item becomes easy.
Check out the following item:
7. Which of the following number lines accurately expresses the range of x if ?
The whole point of this item is to look wacky, because wacky means unsolvable in most people’s minds. The long string of numbers and letters is just two inequalities crammed together. If you like, you could separate them into and . It doesn’t matter. The key thing is to isolate the variable x first. Once you do that, you can jump into the answer choices.
You learned earlier that you can perform any operation on an equation, as long as you perform the same operation to the other side. The same is true of inequalities. It’s also true of two combined inequalities. Watch what we do to the string of letters and numbers. To isolate x, we’ll first add 10 to each part:
Now we have to get rid of that –2 in front of the x. Because we have to divide by –2, don’t forget to switch the signs:
That’s the inequality manipulation part of this item. Now you must match what you’ve discovered with the number line answer choices. Look at the first segment of your answer, . If x is less than -–1, head to the –1 on all the number lines and look for a hollow circle heading left. The circle is hollow because x is not equal to -–1, so –1 can’t be included. The arrow should head to the left because this covers all the numbers less than -–1.
A pass through the answer choices should eliminate everything except D and E. If you’re pressed for time at this point, you can take a guess and move on. If not, all you need to do is look at the second part of your manipulated equations, which states: . That single line under the > is the bottom half of an equals sign. It’s telling you, “x is greater than or equal to 4.” On a number line, this is represented by a filled-in circle on the number item. The answer is D.
The answer choices for this item show ranges of value that a variable can take. Answer choice D shows the different range of values that x can have if . Because there’s a break between the values x could have, it is called a disjointed range. If this range were to go to some range chiropractor and get its joint fixed, it might end up looking like choice A. Because choice A has an upper bound (4) and a lower bound (right up to –1), it is known as a single range.
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