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Intermediate Algebra

OPENSTAX

$Math Studyset 91影视 Explanations$ Math

OER 2017 Edition 路 1346 pages

ISBN: 9780998625720

Chapter 9: Q. 382 (page 977)

In the following exercises, solve each inequality algebraically and write any solution in interval notation.
$- 3 x^{2} - 4 x + 4 鈮� 0$

Short Answer

Expert verified

The solution in interval notation is $(- 鈭�, - 2] 鈭� [\frac{2}{3}, 鈭�)$ .

Step by step solution

01

Step 1. Given information.

The given inequality is:

$- 3 x^{2} - 4 x + 4 鈮� 0$

02

Step 2. Determine the critical points.

Change the inequality sign to an equal sign and then solve the equation.

$- 3 x^{2} - 4 x + 4 = 0$

Split the middle term.

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

The two critical points $- 2$ and $\frac{2}{3}$ divide the number line is three intervals $(- 鈭�, - 2), (- 2, \frac{2}{3}), (\frac{2}{3}, 鈭�)$ .

03

Step 3. Determine the intervals where the inequality is correct.

Test point for the interval $(- 鈭�, - 2)$ is $x = - 3$ .

$- 3 (- 3)^{2} - 4 (- 3) + 4 = - 11 鈮� 0$

Test point for the interval $(- 2, \frac{2}{3})$ is $x = 0$ .

$- 3 (0)^{2} - 4 (0) + 4 = 4 > 0$

Test point for the interval $(\frac{2}{3}, 鈭�)$ is $x = 1$ .

$- 3 (1)^{2} - 4 (1) + 4 = - 3 鈮� 0$

The inequality $- 3 x^{2} - 4 x + 4 鈮� 0$ is true over the intervals $(- 鈭�, - 2]$ and $[\frac{2}{3}, 鈭�)$ .

04

Step 4. Conclusion.

The solution in interval notation is $(- 鈭�, - 2] 鈭� [\frac{2}{3}, 鈭�)$ .

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