Life isn’t always fair. That’s why there are inequalities. An inequality is like an equation, but instead of relating equal quantities, it specifies exactly how two quantities are not equal. There are four types of inequalities:
  1. x > yx is greater than y
  2. x < yx is less than y
  3. x is greater than or equal to y
  4. x is less than or equal to y
Solving inequalities is exactly like solving equations except for one very important difference: When both sides of an inequality are multiplied or divided by a negative number, the direction of the inequality switches.
Here are a few examples:
Solve for x in the inequality – 3 < 2y.
Solve for x in the inequality .
Notice that in the last example the inequality had to be flipped, since both sides had to be divided by –2 between the second and third steps.
To help remember that multiplication or division by a negative number reverses the direction of the inequality, remember that if x > y, then , just as 5 > 4 and –5 < –4. The larger the number, the harder it falls (or the smaller it becomes when you make it negative).
Inequalities are also used to express the range of values that a variable can take. a < x < b means that the value of x is greater than a and less than b. Consider the following word problem:
A company manufactures car parts. As is the case with any system of mass production, small errors in production of every part occur. To make viable car parts, the company must make sure the unavoidable errors occur only within a specific range. The company knows that a particular part they manufacture will not work if it weighs less than 98% of its target weight or more than 102% of its target weight. If the target weight of this piece is 20.5 grams, what is the range of weights the part must fall within for it to function?
The car part must weigh between .98 × 21.5 = 21.07 grams and 1.02 × 21.5 = 21.93 grams. The problem states that the part cannot weigh less than the minimum weight or more than the maximum weight in order for it to work. This means that the part will function at boundary weights themselves, and the lower and upper bounds are included. The answer to the problem is , where x is the weight of the part in grams.
Finding the range of a particular variable is essentially an exercise in close reading. Every time you come across a question involving ranges, you should carefully peruse the problem to pick out whether or not a particular variable’s range includes its bounds. This inclusion is the difference between “less than or equal to”() and simply “less than” (<).
Operations on Ranges
Ranges can be added, subtracted, or multiplied.
If 4 < x < 7, what is the range of 2x + 3?
To solve this problem, simply manipulate the range like an inequality until you have a solution. Begin with the original range:
Multiply the inequality by 2:
Add 3 to the inequality, and you have the answer:
And always remember the crucial rule about multiplying inequalities: If you multiply a range by a negative number, you must flip the greater-than or less-than signs. If you multiply the range 2 < x < 8 by –1, the new range will be –2 > –x > –8.
Absolute Value and Inequalities
Absolute values do the same thing to inequalities that they do to equations. You have to split the inequality into two equations and solve for each. This can result in solutions to inequalities in which the variable falls between two values (as in a range) or a combination of two “disjointed ranges.”
Single Range
If the absolute value is less than a given quantity, then the solution is a single range with a lower and an upper bound. An example of a single range would be the numbers between –5 and 5, as seen in the number line below:
On the SAT, you’ll most likely be asked to deal with single ranges in the following way:
Solve for x in the inequality |2x – 4| ≤ 6.
First, split the inequality into two. Remember to flip around the inequality sign when you write out the inequality for the negative number.
Solve the first:
Then solve the second:
So x is greater than or equal to –1 and less than or equal to 5. In other words, x lies between those two values. So you can write out the value of x in a single range, .
Disjointed Ranges
You won’t always find that the value of the variable lies between two numbers. Instead, you may find that the solution is actually two separate ranges: one whose lower bound is negative infinity and whose upper bound is a real number, and one whose lower bound is a real number and whose upper bound is infinity. Yeah, words make it sound confusing. A number line will make it clearer.
An example of a disjointed range would be all the numbers smaller than –5 and larger than 5, as shown below:
On the SAT, disjointed ranges come up on problems like the following:
Solve for x in the inequality |3x + 4| > 16.
You know the drill. Split ’er up, then solve each part:
Solving the first part:
And the second:
Notice, though, that x is greater than the positive number and smaller than the negative number. In other words, the possible values of x don’t lie between the two numbers, they lie outside the two numbers. So you need two separate ranges to show the possible values of x: –∞ < x < –20/3 and 4 < x < ∞.
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