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Inequalities

Inequalities

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"
xy means "x is less than or equal to y"
x > y means "x is greater than y"
xy means "x is greater than or equal to y"
xy 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 x2≠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}.