Answered: Solve the polynomial inequality 3x2 -…

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Question

Solve the polynomial inequality 3x² - 5x ≤ 0 and graph the solution set on a real number line. Express the solution set in interval notation.

Expert Solution

Step 1

Given that:

The polynomial inequality $3 x^{2} - 5 x \leq 0$

Step 2

To find the solution set in the interval notation.

$L e t, 3 x^{2} - 5 x \leq 0 x (3 x - 5) \leq 0 T o g e t, x \leq 0, (3 x - 5) \leq 0 T h a t i s, x = 0, 3 x - 5 = 0 S o l v e 3 x - 5 = 0 f o r x .$

Add on both side by 5,

To get,

3x = 5

Divide on both side by 3,

To get,

$x = \frac{5}{3} T o g e t, x = 0, x = \frac{5}{3} . T o g e t, x = 0 o r 0 < x < \frac{5}{3} o r x = \frac{5}{3} . T o c h e c k t h e g i v e n i n t e r v a l s s a t i s f y t h e r e q u i r e d c o n d i t i o n \leq 0 . F o r x < 0, T o g e t, t h e v a l u e o f x (3 x - 5) i s p o s i t i v e . F o r x = 0, T o g e t, x (3 x - 5) = 0 . F o r 0 < x < \frac{5}{3}, T o g e t, T h e v a l u e x (3 x - 5) i s n e g a t i v e . F o r x = \frac{5}{3} T o g e t, x (3 x - 5) = 0 . T h u s, F o r t h e v a l u e o f x = 0 o r x = \frac{5}{3} s a t i s f y t h e g i v e n i n e q u a l i t y 3 x^{2} - 5 x \leq 0 . T o g e t, I n t e r v a l N o t a t i o n : [0, \frac{5}{3}]$

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