1. Inequalities and Applications
4.1
Inequalities and Applications
Solving Inequalities
Interval Notation
The Addition Principle for Inequalities
The Multiplication Principle for Inequalities
Using the Principles Together
Problem Solving
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2. Solving Inequalities Examples An inequality is any sentence containing
Any value for a variable that makes an inequality true is called a solution. The set of all solutions is called the solution set. When all solutions of an inequality are found, we say that we have solved the inequality.
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3. Example Solution Determine whether 5 is a solution to
We substitute to get 3(5) + 2 > 7, or 17 >7, a true statement. Thus, 5 is a solution.
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4. The graph of an inequality is a visual representation of the inequality’s solution set. An inequality in one variable can be graphed on a number line.
Example
Graph x < 2 on a number line.
Solution
Note that in set-builder notation the solution is
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5. Interval Notation
Another way to write solutions of an inequality in one variable is to use interval notation. Interval notation uses parentheses, ( ), and brackets, [ ].
If a and b are real numbers such that a < b, we define the open interval (a, b) as the set of all numbers x for which a < x < b. Thus,
a b
(a, b)
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6. Interval Notation
The closed interval [a, b] is defined as the set of all numbers x for which
Thus,
a b
[a, b]
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7. Interval Notation
There are two types of half-open intervals, defined as follows:
(a, b]
a b
a b
[a, b)
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8. Interval Notation
We use the symbols to represent positive and negative infinity, respectively. Thus the notation (a, ) represents the set of all real numbers greater than a, and ( , a) represents the set of all numbers less than a. a
The notations (– , a] and [a, ) are used when we want to include the endpoint a.
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9. The Addition Principle for Inequalities
Two inequalities are equivalent if they have the same solution set. For example, the inequalities x > 3 and 3 < x are equivalent. Just as the addition principle for equations produces equivalent equations, the addition principle for inequalities produces equivalent inequalities.
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10. The Addition Principle for Inequalities
For any real numbers a, b, and c:
a < b is equivalent to a + c < b + c;
a > b is equivalent to a + c > b + c;
Similar statements hold for
Since division by c is the same as multiplication by 1/c, there is no need for a separate division principle.
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11. Example Solution Solve and graph x – 2 > 7. x – 2 > 7
The solution set is
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12. The Multiplication Principle for Inequalities
For any real numbers a, b, and for any positive number c:
a < b is equivalent to ac < bc;
a > b is equivalent to ac > bc.
For any real numbers a, b, and for any negative number c:
a < b is equivalent to ac > bc;
a > b is equivalent to ac < bc.
Similar statements hold for
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13. Example Solution Solve and graph – 2x > 6. – 2x > 6 – 2x < 6
The symbol must be reversed.
– 2x < 6
– – 2
x < – 3
The solution set is
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14. Using the Principles Together
Example
Solve and graph
Solution
The solution set is
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15. Problem Solving
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