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:
x > y — x is
greater than y
< y — x is less
- — x is
greater than or equal to y
- — 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:
for x in the inequality – 3 < 2y.
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
, just as 5
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
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
The car part must weigh between .98 × 21.5 = 21.07
and 1.02 × 21.5 = 21.93
grams. The problem
states that the part cannot weigh less
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
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.
< x < 7, what is the range of 2x +
To solve this problem, simply manipulate the range like
an inequality until you have a solution. Begin with the original
Multiply the inequality by 2:
Add 3 to the inequality, and you have the
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.”
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:
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:
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
a single range,
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
On the SAT, disjointed ranges come up on problems like
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 < ∞.