Section 3-1 Linear Inequalities; Absolute Value

Inequalities Inequalities can be written in one or more variables. Linear Inequalities: 2x + 3y > 6 Polynomial Inequalities:

Solving Inequalities The properties used in solving linear inequalities are similar to the properties used in solving linear equations. You can add the same number to (or subtract the same number from) both sides of an inequality. That is, if a < b, then a + c < b + c

Solving Inequalities You can multiply (or divide) both sides of an inequality by the same POSITIVE number. That is, if a 0, then ac < bc. You can multiply (or divide) both sides of an inequality by the same NEGATIVE number if you reverse the inequality sign. That is, if a bc.

To graph inequalities: If an endpoint is to be included in the graph, a solid dot (●) is used. Use a solid dot when the inequality symbol is, or =. An open dot (○) indicates that the endpoint is not to be included in the graph. Use an open dot when the inequality symbol is.

Note… ** When inequalities are combined, sometimes the combination represents the empty set. For example: the statement 6 < x < 1 says that 6 < x and x < 1. A number cannot be greater than 6 AND less than 1 at the same time. Since no real number x satisfies both these inequalities, the combined inequality represents the empty set (in other words, no solution)

Examples Solve and graph: X – 8 < -12 ½y > -2 1 – 2d < 7

Examples Solve and graph: -7x – 1 > 13

Absolute Value The absolute value of a number x, denoted can be interpreted geometrically as the distance from x to zero in either direction on the number line. You can solve linear equations and inequalities involving. Since c represents a distance so c > 0.

Sentences involving SentenceMeaningGraphSolution = c The distance from x to 0 is exactly c units X = c or x = -c < c The distance from x to 0 is less than c units -c < x < c > c The distance from x to 0 is greater than c units X c

Equations and inequalities involving, where k is a constant, can be solved by interpreting as the distance from x to k on the number line.

Absolute value sentences can be solved algebraically by using the three sets of equivalent sentences listed below. SentenceEquivalent Sentence = c ax + b = ±c < c -c < ax + b < c > c ax + b c

Examples Solve and graph.