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What is an Inequality?

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

*x* < *y* means "*x* is less than *y*"

*x*≤*y* means "*x* is less than or equal to *y*"

*x* > *y* means "*x* is greater than *y*"

*x*≥*y* means "*x* is greater than or equal to *y*"

*x*≠*y* means "*x* is **not** equal to *y*"

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

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Inequalities with Variables

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

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Finding a Solution Set from a Replacement Set

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}.