Inequalities: Inequalities

Inequalities

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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

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}.