To determine To calculate: The values of x which satisfy the inequality f ( x ) ≤ 5 if, the function is f ( x ) = 2 x 2 + 9 x − 1 and represent the solution on the number line. Explanation Given Information: The provided inequality is f ( x ) ≤ 5 and the function is f ( x ) = 2 x 2 + 9 x − 1 . Formula used: Follow the steps as mentioned below to solve the quadratic, polynomial and rational inequalities. Step1: Compute the boundary values by solving the inequality for x via treating inequality as equation. Step2: Draw a number line for the solution obtained from step 1 by using following points, Use open circle ∘ for the inequality < or > . Use closed circle • for the inequality ≤ or ≥ . Step3: For a rational inequality compute the boundary values by equating the denominator equal to zero, represent these boundary conditions on number line by open circle ∘ . Step4: From each interval check the inequality at any test point within the interval. Step5: The set of points that satisfy the inequality will be the solution. Calculation: Consider the inequality f ( x ) ≤ 5 Also consider the function, f ( x ) = 2 x 2 + 9 x − 1 Substitute the function in the inequality, 2 x 2 + 9 x − 1 ≤ 5 Treat the inequality as equation, 2 x 2 + 9 x − 1 = 5 Take the factors, 2 x 2 + 9 x − 1 − 5 = 0 2 x 2 + 9 x − 6 = 0 If the equation is in the form a x 2 + b x + c = 0 provided ( a ≠ 0 ) then, according to quadratic formula, the solution of the equation is x = − b ± b 2 − 4 a c 2 a . Compute the values of x by using the formula, x = − 9 ± 9 2 − 4 ( 2 ) ( − 6 ) 2 ⋅ 2 = − 9 ± 81 + 48 4 = − 9 ± 129 4 Therefore, the boundary values for the quadratic inequality are − 9 + 129 4 and − 9 − 129 4 . Here the inequality is ≤ , so, use close circle to represent the solution and the boundary values on the number line, as, Form the table to check the test points with in the interval for all the possible intervals, Interval A Interval B Interval C ( − ∞ , − 9 − 129 4 ] [ − 9 − 129 4 , − 9 + 129 4 ] [ − 9 + 129 4 , ∞ ) Test value − 10 Test value − 1 Test value 1 2 x 2 + 9 x − 1 ≤ 5 2 ( − 10 ) 2 + 9 ( − 10 ) − 1 ≤ ? 5 200 − 90 − 1 ≤ 5 ? 109 ≤ 5 This is not possible, so, this interval is false

Intermediate Algebra For College Students (10th Edition)

10th Edition

ISBN: 9780134758992

Author: Allen R. Angel, Dennis Runde

Publisher: PEARSON

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Chapter 8.6, Problem 39ES

To determine

To calculate: The values of x which satisfy the inequality f(x)≤5 if, the function is f(x)=2x2+9x−1 and represent the solution on the number line.

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Chapter 8.6, Problem 38ES

Chapter 8.6, Problem 40ES

Chapter 8 Solutions

Intermediate Algebra For College Students (10th Edition)

Ch. 8.1 - Fill in the blanks with the appropriate word,...Ch. 8.1 - Fill in the blanks with the appropriate word,...Ch. 8.1 - Prob. 3ES Ch. 8.1 - Prob. 4ES Ch. 8.1 - Fill in the blanks with the appropriate word,...Ch. 8.1 - Prob. 6ES Ch. 8.1 - Prob. 7ES Ch. 8.1 - Prob. 8ES Ch. 8.1 - Prob. 9ES Ch. 8.1 - Prob. 10ES

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