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MTH 070 Elementary Algebra Chapter 2 Equations and Inequalities in One Variable with Applications 2.6 – Absolute Value Equations and Inequalities Copyright © 2010 by Ron Wallace, all rights reserved. Absolute Value x The unsigned distance a quantity is from zero. 17 8 0 Absolute Value x The unsigned distance a quantity is from zero. x if x 0 x x if x 0 Absolute Value Application How do you find the distance between two values? Examples: 7 & 12 –3 & –15 8 & –11 –4 & 9 Pattern? Subtract the numbers & ignore the sign of the result. Absolute Value Application How do you find the distance between two expressions? Example: 2x & 3+x Subtract them? But in which order to guarantee the result is positive? Consider with x = 4 and x = 2 Absolute Value – One more thing. x a What can be said about the value of a? Absolute Value Equations x 7 What can be said about the value of x? Solving Absolute Value Equations expression1 expression2 How can this be rewritten without the absolute value notation? Solving Absolute Value Equations Strategy Get the absolute value on one side of the equation and everything else on the other side. Simplify if possible. Rewrite without the absolute value notation. Gives 2 equations related by the word “or” Solve the resulting two equations. Check. Solving Absolute Value Equations Examples 2x 1 7 x 5 8 2 x 3 2x 4 Absolute Value Inequalities w/ “less than” Quality control requires that the diameter of a particular hole must be within 0.01” of 2”. If x represents the diameter, how could you express the value of x algebraically. x 2 0.01 Absolute Value Inequalities w/ “less than” Find some values that would be solutions to the following … (graph them on a number line – notice anything?) x 2 Absolute Value Inequalities w/ “less than” General result ... x a a x a Therefore, to solve … Rewrite as a double inequality Solve Check (at and around the endpoints) Absolute Value Inequalities w/ “less than” Examples 6 x 18 x5 2 3x 4 7 Absolute Value Inequalities w/ “greater than” Find some values that would be solutions to the following … (graph them on a number line – notice anything?) x 3 Absolute Value Inequalities w/ “greater than” General result ... x a x a or x a Therefore, to solve … Rewrite as a compound inequality w/ “or” Solve Check (at and around the endpoints) Absolute Value Inequalities w/ “less than” Examples 6 x 18 x5 2 3x 4 7