Copyright 2013, 2010, 2007, Pearson, Education, Inc. Section 6.6 Linear Inequalitie s
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Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Section 6.6
Linear Inequalitie
s
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
What You Will Learn
Solving Linear InequalitiesSolving Compound Inequalities
6.6-2
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Symbols of Inequality
a < b means that a is less than b.
a ≤ b means that a is less than or equal to b.
a > b means that a is greater than b.
a ≥ b means that a is greater than or equal to b.
6.6-3
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Inequality
An inequality consists of two (or more) expressions joined by an inequality sign.
3 < 5, x < 2, 3x −2 ≥5
6.6-4
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Solving InequalitiesTo indicate the solution set of x < 2 on the number line, we draw an open circle at 2 and a line to the left of 2 with an arrow at its end. The open circle indicates that the solution set does not include the number 2.
6.6-5
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Solving InequalitiesTo indicate the solution set of x ≤ 2 on the number line, we draw a closed circle at 2 and a line to the left of 2 with an arrow at its end. The closed circle indicates that the solution set does not include the number 2.
6.6-6
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Solving InequalitiesFind the solution to an inequality by adding, subtracting, multiplying or dividing both sides by the same number or expression.Reverse the direction of the inequality symbol when multiplying or dividing both sides of an inequality by a negative number.
6.6-7
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Example 4: Dividing by a Negative NumberSolve the inequality –5x < 20 and graph the solution set on the number line.Solution
−5 x < 20
−5 x−5
>20−5
x > −4
6.6-8
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Compound Inequality
An inequality of the form a < x < b is called a compound inequality.
It means that a < x and x < b.
6.6-9
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Example 7: A Compound InequalityGraph the solution set of the inequality –3 < x ≤ 2 where x is an integer.
SolutionThe solution set is the integers between –3 and 2, including 2.
6.6-10
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Example 7: A Compound InequalityGraph the solution set of the inequality –3 < x ≤ 2 where x is a real number.
SolutionThe solution set is all real numbers between –3 and 2, including 2.
6.6-11
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Example 9: Average Grade
A student must have an average (the mean) on five tests that is greater than or equal to 80% but less than 90% to receive a final grade of B. Devon’s grades on the first four tests were 98%, 76%, 86%, and 92%. What range of scores on the fifth test will give him a B in the course?
6.6-12
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Example 9: Average GradeSolutionLet x = the fifth grade
Average =
98 + 76 + 86 + 92 + x5
80 ≤
98 + 76 + 86 + 92 + x5
< 90
80 ≤
352 + x5
< 90
6.6-13
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Example 9: Average GradeSolution
5 80( ) ≤ 5
352 + x
5
⎛
⎝⎜
⎞
⎠⎟ < 5 90( )
400 ≤352 + x < 450 400 −352 ≤352 −352 + x < 450 −352
48 ≤x < 98A grade of 48% up to but not including a grade of 98% on the fifth test will result in a grade of B in this course.6.6-14
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