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Q. 21

Expert-verified

Precalculus Enhanced with Graphing Utilities

Found in: Page 171

Book edition 6th

Author(s) Sullivan

Pages 1200 pages

ISBN 9780321795465

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Short Answer

In Problems 7–22, solve each inequality.
$6 (x^{2} - 1) > 5 x$

The solution set is $\{x : x < \frac{- 2}{3} or x > \frac{3}{2}\}$ .

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Given Information

The given inequality is $6 (x^{2} - 1) > 5 x$ .

Step 2. Find the Intercepts and Vertex

Express the inequality in standard form.

$6 x^{2} - 5 x - 6 > 0$

Factor the function $f (x) = 6 x^{2} - 5 x - 6$ .

$f (x) = (2 x - 3) (3 x + 2)$

So, the x-intercept of the function are $(\frac{3}{2}, 0), (- \frac{2}{3}, 0)$ .
The value of $f (0) = - 6$ .
So, the y-intercept is $(0, - 6)$ .
Find x -coordinate of the vertex by using the formula $x = \frac{- b}{2 a}$ .

$x = \frac{5}{12}$

The value of $f (\frac{5}{12}) = \frac{- 169}{24}$ .
So, the vertex of the parabola is $(\frac{5}{12}, \frac{- 169}{24})$ ..

Step 3. Plot the Function

Plot the curve of the function on the graph by using the obtained vertices and intercepts.

Step 4. Find the region above x-axis

From the graph, it is observed that the curve is above x-axis when $x < \frac{- 2}{3} or x > \frac{3}{2}$ .
So, the solution set is $\{x : x < \frac{- 2}{3} or x > \frac{3}{2}\}$

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