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Unformatted text preview: Section 1.7 Inequalities Linear Inequalities An inequality is linear if each term is constant or a multiple of the variable. EXAMPLE: Solve the inequality 3 x < 9 x + 4 and sketch the solution set. Solution: We have 3 x < 9 x + 4 3 x 9 x < 9 x + 4 9 x 6 x < 4 ( 1 6 ) ( 6 x ) > ( 1 6 ) (4) x > 2 3 The solution set consists of all numbers greater than 2 3 . In other words the solution of the inequality is the interval ( 2 3 , ) . EXAMPLE: Solve the inequalities 4 3 x 2 < 13 and sketch the solution set. 1 EXAMPLE: Solve the inequalities 4 3 x 2 < 13 and sketch the solution set. Solution: We have 4 3 x 2 < 13 4 + 2 3 x 2 + 2 < 13 + 2 6 3 x < 15 ( 1 3 ) (6) ( 1 3 ) (3 x ) < ( 1 3 ) (15) 2 x < 5 Therefore, the solution set is [2 , 5) . EXAMPLE: Solve the inequalities 4 < 5 3 x 17 and sketch the solution set. Solution: We have 4 < 5 3 x 17 4 5 < 5 3 x 5 17 5 9 < 3 x 12 ( 1 3 ) (12) ( 1 3 ) ( 3 x ) < ( 1 3 ) ( 9) 4 x < 3 Therefore, the solution set is [ 4 , 3) . 2 Nonlinear Inequalities EXAMPLE: Solve the inequality x 2 5 x 6 and sketch the solution set. Solution: The corresponding equation x 2 5 x + 6 = ( x 2)( x 3) = 0 has the solutions 2 and 3 . As shown in the Figure below, the numbers 2 and 3 divide the real line into three intervals: ( , 2) , (2 , 3) , and (3 , ). 2 3 On each of these intervals we determine the signs of the factors using test values . We choose a number inside each interval and check the sign of the factors x 2 and x 3 at the value selected. For instance, if we use the test value x = 1 from the interval ( , 2) shown in Figure above, then substitution in the factors...
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This note was uploaded on 01/23/2012 for the course MATH 8650 taught by Professor Kiryltsishchanka during the Spring '12 term at NYU.
 Spring '12
 KIRYLTSISHCHANKA
 Calculus, Algebra, Inequalities

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