Inequalities
Inequalities
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
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
5x + 3≤9
true for
x = 1
?
5(1) + 3≤9
?
8≤9
? Yes.
Thus,
5x + 3≤9
is true for
x = 1
.
Example 2. Does
3x - 2 > 2x + 1
hold for
x = 3
?
3(3) - 2 > 2(3) + 1
?
7 > 7
? No.
Thus,
3x - 2 > 2x + 1
does not hold for
x = 3
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
-3x≥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≠2x
from the
replacement set
{0, 1, 2, 3}
.
02≠2(0)
? False (they are, in fact, equal).
12≠2(1)
? True.
22≠2(2)
? False.
32≠2(3)
? True.
Thus, the solution set is
{1, 3}
.
