An inequality is a statement that shows the relationship between two (or more)
expressions with one of the following five signs:
<
,
≤
,
>
,
≥
,
≠
.

x<ymeans "xis less thany"

x≤ymeans "xis less than or equal toy"

x>ymeans "xis greater thany"

x≥ymeans "xis greater than or equal toy"

x≠ymeans "xisnotequal toy"

Like an equation, an inequality can be true or false.

34 - 12 > 5 + 2 is a *true* statement.

1 + 3 < 6 - 2 is a *false* statement.

1 + 3≤6 - 2
is a *true* statement.

1 + 3≠6 - 2
is a *false* statement.

-20 < - 18
is a *true* statement

To determine whether an inequality is true or false for a given value of a variable, plug in
the value for the variable. If an inequality is true for a given value, we say that it
*holds* for that value.

*Example 1.* Is
5*x* + 3≤9
true for
*x* = 1
?

5(1) + 3≤9
?

8≤9
? Yes.

Thus,
5*x* + 3≤9
is true for
*x* = 1
.

*Example 2.* Does
3*x* - 2 > 2*x* + 1
hold for
*x* = 3
?

3(3) - 2 > 2(3) + 1
?

7 > 7
? No.

Thus,
3*x* - 2 > 2*x* + 1
does not hold for
*x* = 3

Finding a solution set to an inequality, given a replacement set, is similar to finding a solution set to an equation. Plug each of the values in the replacement set in for the variable. If the inequality is true for a certain value, that value belongs in the solution set.

*Example 1*: Find the solution set of
*x* - 5 > 12
from the
replacement set
{10, 15, 20, 25}
.

10 - 5 > 12 ? False.

15 - 5 > 12 ? False.

20 - 5 > 12 ? True.

25 - 5 > 12 ? True.

Thus, the solution set is
{20, 25}
.

*Example 2*: Find the solution set of
-3*x*≥6
from the
replacement set
{ -4, -3, -2, -1, 0, 1}
.

-3(- 4)≥6
? True.

-3(- 3)≥6
? True.

-3(- 2)≥6
? True.

-3(- 1)≥6
? False.

-3(0)≥6
? False.

-3(1)≥6
? False.

Thus, the solution set is
{ -4, -3, -2}
.

*Example 3.* Find the solution set of
*x*
^{2}≠2*x*
from the
replacement set
{0, 1, 2, 3}
.

0^{2}≠2(0)
? False (they are, in fact, equal).

1^{2}≠2(1)
? True.

2^{2}≠2(2)
? False.

3^{2}≠2(3)
? True.

Thus, the solution set is
{1, 3}
.

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