Q. 16 In problems 7-22, solve the ineq... [FREE SOLUTION]

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Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 3: Q. 16 (page 171)

In problems 7-22, solve the inequality.
$6 x^{2} < 6 + 5 x$

Short Answer

Expert verified

The solution set is $(\frac{- 2}{3}, \frac{3}{2})$ .

Step by step solution

Step 1. Given Information

The given inequality is $6 x^{2} < 6 + 5 x$ .

Step 2. Find the intercepts and vertex

Rewrite the given inequality.

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

The value of $f (0) = - 6$ .
So, the y-intercept is $(0, - 6)$ .
Factor the equation.

$\begin{array}{rcl} 6 x^{2} - 5 x - 6 & = & 0 \\ 6 x^{2} - 9 x + 4 x - 6 & = & 0 \\ 3 x (2 x - 3) + 2 (2 x - 3) & = & 0 \\ (3 x + 2) (2 x - 3) & = & 0 \end{array}$

Equate the factors to 0.

$\begin{array}{rcl} 3 x + 2 & = & 0 \\ x & = & \frac{- 2}{3} \\ 2 x - 3 & = & 0 \\ x & = & \frac{3}{2} \end{array}$

So, the x-intercepts are $(\frac{- 2}{3}, 0), (\frac{3}{2}, 0)$ .
The vertex of the parabola exists at $\frac{- b}{2 a} = \frac{- (- 5)}{2 (6)} = \frac{5}{12}$ .
Find the value of the function at $x = \frac{5}{12}$ .

$\begin{array}{rcl} f (\frac{5}{12}) & = & 6 {(\frac{5}{12})}^{2} - 5 (\frac{5}{12}) - 6 \\ = & 6 (\frac{25}{144}) - \frac{25}{12} - 6 \\ = & \frac{150 - 300 - 864}{144} \\ = & \frac{1014}{144} \\ = & \frac{169}{24} \end{array}$

So, the vertex is $(\frac{5}{12}, \begin{array}{rcl} \frac{169}{24} \end{array})$ .

Step 3. Plot the graph

Plot the parabola on a graph using the vertex and intercepts calculated in the previous step.

Step 4. Find the solution set

From the graph, it is observed that for x belonging to $(\frac{- 2}{3}, \frac{3}{2})$ , $y < 0$ .
So, the solution set of the given inequality is $(\frac{- 2}{3}, \frac{3}{2})$ .

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91影视

In problems 7-22, solve the inequality.
$6 x^{2} < 6 + 5 x$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Find the intercepts and vertex

Step 3. Plot the graph

Step 4. Find the solution set

One App. One Place for Learning.

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91影视

In problems 7-22, solve the inequality. 6x2<6+5x

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Find the intercepts and vertex

Step 3. Plot the graph

Step 4. Find the solution set

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Logic and Functions

Mechanics Maths

Theoretical and Mathematical Physics

Statistics

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

In problems 7-22, solve the inequality.
$6 x^{2} < 6 + 5 x$