Solving Inequalities Solving Inequalities • Solve Inequalities ≡ find value(s) of the variable(s) that make the inequality true • Inequalities almost always generate an infinite number of solutions (a set or interval of numbers on the R Number line) • Testing answer(s) is more complex: • You can’t test every one • Test boundaries w/ corresponding equation • Test points in each interval (sign graph) 3 Ways to Represent Solutions Compound: or F either, union 1. Inequality notation uses <, , >, : and F both, intersection x4 x0 x 1 or x4 3 x 7 x 3 and x 7 unbounded bounded • Interval notation uses (a,b), [a,b], (a,b], [a,b): – a and b are the endpoints of the interval and a b – If an endpoint is included, a square bracket is used: [ or ] is not included, a parenthesis is used: ( or ) is ±∞, it is never included (±∞ Real numbers) [4, ) (,0) (,1) [4, ) (3,7] (,7] (3, ) unbounded bounded 3. Graph on the number line: ex: x 4 or [4, ∞) 0 4 Try these: Write in inequality and interval notation and graph on number line. 1. x is greater than -5. 2. y is at most 24. 3. x is greater than 2 and at most 6 4. x is less than or equal to -1 or greater than 16 Weird or Boundary cases • Open-Open Bounded Interval Notation can be mistaken for a point: Is (3,4) an ordered pair representing (x,y) or f (3) 4 [function notation] ? or Is (3,4) the open interval from 3 to 4, i.e., 3 < x < 4? Context is important • Compound inequalities (“or” or “and”) can lead to solutions that are: all Reals: or the empty set: Notice the violation of x 0 or x 3 4 x2 the Transitive Property! x x 4 and x 2 Strategies to Solve Inequalities Type Strategy Linear Isolate the variable by simplifying and using the Addition and Multiplication Properties of Inequality. (~linear equalities) *** Reverse the direction of the inequality symbol if you multiply or divide by a negative quantity. *** Absolute Value Isolate the absolute value and then use the definition of absolute value to create two cases. (~absolute value equalities). If , then join by “or”. If , then join by “and”. Other types: Find critical numbers where the expression changes sign or is undefined. Sign changes are found by converting to standard form and solving the corresponding equality (= 0). The critical numbers create intervals on the number line to be tested in the original inequality. Polynomial Find critical numbers (solve corresponding equation). (quadratic or Test values w/in each interval to determine the intervals that make the higher degree) original inequality true (sign graph). Rational Place in standard form and combine fractions into one rational expression (do not clear fractions as you would for Equalities). Find critical numbers: where numerator is 0 (sign change) and where denominator is 0 (undefined). Test values w/in each interval to determine the intervals that make the original inequality true (sign graph).
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