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Copyright © 2011 Pearson Education, Inc.
Linear and Absolute Value Inequalities
Section 1.7
Equations, Inequalities, and Modeling
Copyright © 2011 Pearson Education, Inc. Slide 1-3
1.7
An equation states that two algebraic expressions are equal, while an inequality or simple inequality is a statement that two algebraic expressions are not equal in a particular way.
The solution set to an inequality is the set of all real numbers for which the inequality is true. It is noted as an interval, and is written in interval notation.
Intervals that use the infinity symbol are unbounded intervals.
An unbounded interval with an endpoint is open if the endpoint is not included in the interval and closed if the endpoint is included.
Interval Notation
Copyright © 2011 Pearson Education, Inc. Slide 1-4
1.7Interval Notation forUnbounded Intervals
Copyright © 2011 Pearson Education, Inc. Slide 1-5
1.7
Replacing the equal sign in the general linear equation ax + b = 0 by any of the symbols <, ≤, >, or ≥ gives a linear inequality.
Two inequalities are equivalent if they have the same solution set.
When an inequality is multiplied or divided by a negative number, the direction of the inequality symbol is reversed.
Multiplication and division with a variable expression is usually avoided because we do not know whether the expression is positive or negative.
Linear Inequalities
Copyright © 2011 Pearson Education, Inc. Slide 1-6
1.7
Properties of InequalityIf A and B are algebraic expressions and C is a nonzero real number, then the inequality A < B is equivalent to
1. 2.
3.
,CBCA ,, negative for positive for CCBCACCBCA
.negative for ,positive for CC
B
C
AC
C
B
C
A
Linear Inequalities
Copyright © 2011 Pearson Education, Inc. Slide 1-7
1.7
A compound inequality is a sentence containing two simple inequalities connected with “and” or “or.”
The solution to a compound inequality can be an interval of real numbers that does not involve infinity—a bounded interval of real numbers.
Because a bounded interval contains both of its endpoints, the interval is closed.
The solution set to a compound inequality using the connector “or” is the union of two solution sets, and the solution set to a compound inequality using the “and” is the intersection of the two solutions sets.
Compound Inequalities
Copyright © 2011 Pearson Education, Inc. Slide 1-8
1.7
Interval Notation for Bounded Intervals
Copyright © 2011 Pearson Education, Inc. Slide 1-9
1.7
Absolute Value Inequalities
