322 MCGRAW-HILL’S SAT
Inequalities as Unbalanced Scales
Inequalities are just unbalanced scales. Nearly
all of the laws of equality pertain to inequalities,
with one exception. When solving inequalities,
keep the direction of the inequality (remember
that “the alligator < always eats the bigger num-
ber”) unless you divide or multiply by a nega-
tive, in which case you “switch” the inequality.Example:
Solve x^2 > 6xfor x.
You might be tempted to divide both sides by xand
get x> 6, but this incorrectly assumes that xis posi-
tive. If xis positive, then x> 6, but if xis negative, then
x< 6. (Switch the inequality when you divide by a
negative!) But of course anynegative number is less
than 6, so the solution is either x> 6 or x< 0. (Plug in
numbers to verify!)
Absolute Values as Distances
The absolute value of x, written as ⏐x⏐, means
the distance from xto 0 on the number line.
Since distances are never negative, neither are
absolute values. For instance, since −4 is four
units away from 0, we say ⏐− 4 ⏐=4.The distance between numbers is found from
their difference. For instance, the distance be-
tween 5 and −2 on the number line is 5 −(−2) =7.
But differences can be negative, and distances
can’t! That’s where absolute values come in.
Mathematically, the distance between a and b is
a−b.Example:
Graph the solution of ⏐x+ 2 ⏐ ≥3.
You can think about this in two ways. First think about
distances. ⏐x+ 2 ⏐is the same as ⏐x−(−2)⏐, which is the
distance between xand −2. So if this distance must be
greater than or equal to 3, you can just visualize those
numbers that are at least 3 units away from −2:
Or you can do it more “algebraically” if you prefer.
The only numbers that have an absolute value greater
than or equal to 3 are numbers greater than or equal
to 3 or less than or equal to −3, right? Therefore, say-
ing ⏐x+ 2 ⏐≥3 is the same as saying x+ 2 ≥3 or x+ 2
≤−3. Subtracting 2 from both sides of both inequali-
ties gives x≥1 or x≤−5, which confirms the answer
by the other method.Plugging In
After solving each of the examples above, you should, as
with all equations and inequalities, plug inyour solution
to confirm that it works in the equation or inequality.
But plugging in can also be a good way of solving
multiple-choice problems that ask you to find an ex-
pression with variables rather than a numerical solution.If a multiple-choice question has choices that
contain unknowns, you can often simplify the
problem by just plugging in values for the un-
knowns. But think first: in some situations,
plugging in is not the simplest method.Example:
If y=r−6 and z=r+5, which of the following ex-
presses rin terms of yand z?(A) y+z− 1
(B) y+z
(C) y+z+ 1(D)(E)
If you pick rto be 6—it can be whatever you want,
so pick an easy number!—then yis 6 − 6 =0 and zis
6 + 5 =11. The question is asking for an expression for
r,so look for 6 among the choices. Plugging in your
values gives (A) 10 (B) 11 (C) 12 (D) 5 (E) 6. Always
evaluate allthe choices because you must work by
process of elimination.Only (E) gives 6, so it must
be the right answer!yz++ 1
2yz+− 1
2Lesson 6:
Inequalities, Absolute Values, and Plugging In
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