1­4 Inequalities
August 27, 2008
1-4 Solving Inequalities
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Objective:
To solve and graph inequalities
To solve and graph compound inequalities
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1­4 Inequalities
August 27, 2008
Check Skills You'll Need
State whether each inequality is true or false.
1. 5 < 12
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4. 5 < ­12
2. 5 < ­12
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5. 5 < 5
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3. 5 > 12
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6. 5 > 5
Solve each equation.
7. 3x + 3 = 2x ­ 3
8. 5x = 9(x ­ 8) + 12
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1­4 Inequalities
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Properties of Inequalities
Let a, b, and c represent real numbers
Transitive Property
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If a < b and b < c, then a < c
Addition Property
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If a < b, then a + c < b + c
Subtraction Property
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If a < b, then a ­ c < b ­ c
Multiplication Property
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If a < b and c > 0, then ac < bc
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If a < b and c < 0, then ac > bc
Division Property
You must reverse the inequality symbol when c is negative
a_ _ _
b
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If a < b and c > 0, then < c c
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a _ _b
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If a < b and c < 0, then > c c
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1­4 Inequalities
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Solving and Graphing Inequalities
Solve each inequality. Graph the solution.
­2x < 3(x ­ 5)
­2x < 3x ­ 15
­5x < ­15
x > 3
­5 ­4 ­3 ­2 ­1 0 1 2 3 4 5
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1­4 Inequalities
August 27, 2008
No solutions or all real numbers as solutions
Solve each inequality. Graph the solution.
7x > 7(2 + x)
7x > 14 + 7x
0 > 14
0 > 14
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Therefore there is no solution
­5 ­4 ­3 ­2 ­1 0 1 2 3 4 5
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1­4 Inequalities
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1­4 Inequalities
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Real ­ World Connection
A real estate agent earns a salary of $2000 per month plus 4% of the sales. Find the sales if the salesperson is to have a monthly income of at least $5000. Relate:
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$2000 + 4% of sales > $5000
Define:
Let x = monthly real estate sales
Write:
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2000 + .04x > 5000
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.04x > 3000
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x > 75000
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1­4 Inequalities
August 27, 2008
Compound Inequality
A compound inequality is a pair of inequalities joined by and or or.
Examples:
­1 < x and x < 3, which you can also write as ­1 < x < 3
x < ­1 or x > 3
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1­4 Inequalities
August 27, 2008
Compound inequality containing and
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Graph the solution of 2x ­ 1 < 3x and x > 4x ­ 9
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2x ­ 1 < 3x
and
x > 4x ­ 9
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­ 1 < x
­3x > ­9
x < 3
­10 ­9
­8 ­7
­6
­5 ­4
­3
­2
­1
0
1
2
3
4
5
6
7
8
9
10
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1­4 Inequalities
August 27, 2008
Compound inequality containing or
Graph the solution of 3x + 9 < ­3 or ­2x + 1 < 5
or
3x + 9 < ­3
­2x + 1 < 5
­2x < 4
3x < ­12
x > ­2
x < ­4
­10 ­9
­8 ­7
­6
­5 ­4
­3
­2
­1
0
1
2
3
4
5
6
7
8
9
10
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1­4 Inequalities
August 27, 2008
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A strip of wood is to be 17 cm long with a tolerance of 0.15 cm. How much should be trimmed from a strip 18 cm long to allow it to meet specifications?
Relate: _
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minimum length < final length < maximum length
Define:
Let x = number of centimeters to remove
Write:
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17 ­ 0.15 < 18 ­ x < 17 + 0.15
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16.85 < 18 ­ x < 17.15
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­1.15 < ­ x < ­0.85
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1.15 > x > 0.85
You can trim at least 0.85 cm and no more than 1.15 cm.
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1­4 Inequalities
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What is one important difference between solving equations and solving inequalities?
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1­4 Inequalities
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Homework
pages: 29 ­ 30 #'s: 2 ­ 28 even, 29 ­ 32, 36 ­ 37, 42 ­ 54 even
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