HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Section A.4.

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HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Section A.4 Absolute Value Equations and Inequalities

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Solving Absolute Value Equations Solve the following equations involving absolute value. Solution x = −5orx = 5

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Solving Absolute Value Equations Solution There is no number that has a negative absolute value. Therefore, this equation has no solution. (The solution is  and the equation is a contradiction.)

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Absolute Value Inequalities Solving Absolute Value Inequalities with < (or ≤ ) For c > 0: a. If then −c < x < c. b. If then −c < ax + b < c. The inequalities in a. and b. are also true if < is replaced by ≤.

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Solving Absolute Value Inequalities Solve the following absolute value inequalities and graph the solution sets.

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Solving Absolute Value Inequalities

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Solving Absolute Value Inequalities

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Solving Absolute Value Inequalities

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Solving Absolute Value Inequalities

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Absolute Value Inequalities Notes The expression x > 3 or x < −3 cannot be combined into one inequality expression. The word or must separate the inequalities since any number that satisfies one or the other is a solution to the absolute value inequality. There are no numbers that satisfy both inequalities.

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Absolute Value Inequalities Solving Absolute Value Inequalities with > (or ≥ ) For c > 0: a. If then x c. b. If then ax + b c. The inequalities in a. and b. are also true if > is replaced by ≥.

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Solving Absolute Value Inequalities Solve the following absolute value inequalities and graph the solution set.

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Solving Absolute Value Inequalities

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Solving Absolute Value Inequalities There is nothing to do here except observe that no matter what is substituted for x, the absolute value will be greater than −6. Absolute value is always nonnegative (greater than or equal to 0). The solution to the inequality is all real numbers, so shade the entire number line. In interval notation, x is in (−∞, ∞).

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Solving Absolute Value Inequalities

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Section A.6 Graphing Systems of Linear Inequalities

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Solving Systems of Linear Inequalities Graphically 1.For each inequality, graph the boundary line and shade the appropriate half-plane. 2.Determine the region of the graph that is common to both half-planes (the region where the shading overlaps). (This region is called the intersection of the two half-planes and is the solution set of the system.) 3.To check, pick one test-point in the intersection and verify that it satisfies both inequalities. Note: If there is no intersection, then the system has no solution.

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Graphing Systems of Linear Inequalities Graph the points that satisfy the system of inequalities:

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Solve the system of linear inequalities graphically:

HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Solve the system of linear inequalities graphically: