Section_1.7-Inequalities

Section_1.7-Inequalities - Section 1.7 Inequalities Linear...

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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.

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Section_1.7-Inequalities - Section 1.7 Inequalities Linear...

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