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Intermediate Algebra

OPENSTAX

$Math Studyset 91影视 Explanations$ Math

OER 2017 Edition 路 1346 pages

ISBN: 9780998625720

Chapter 2: Q. 424 (page 196)

Solve the inequality, graph the solution on the number line, and write the solution in interval notation:
$- 2 (3 x - 4) 鈮� 2$ and
$- 4 (x - 1) < 鈥� 2$

Short Answer

Expert verified

The solution of the inequality: $x 鈮� 1$ and $x > \frac{1}{2}$

Interval notation: $[1, 鈭�)$

The graph of the inequality on number line is:

Step by step solution

01

Step 1. Given Information.

We have:

$- 2 (3 x - 4) 鈮� 2$ and

$- 4 (x - 1) < 鈥� 2$

02

Step 2. Solve the inequality.

First, solve $- 2 (3 x - 4) 鈮� 2$ :

Multiply both sides by $- 1$ (reverse the inequality):

$(- 2 (3 x - 4)) (- 1) 鈮� 鈥� 2 (- 1) 2 (3 x - 4) 鈮� 鈥� - 2 \frac{2 (3 x - 4)}{2} 鈮� \frac{- 2}{2} 3 x - 4 鈮� 鈥� - 1 3 x - 4 + 4 鈮� 鈥� - 1 + 4 3 x 鈮� 鈥� 3 \frac{3 x}{3} 鈮� \frac{3}{3} x 鈮� 鈥� 1$

Now, solve $- 4 (x - 1) < 鈥� 2$ :

Multiply both sides by $- 1$ (reverse the inequality):

$(- 4 (x - 1)) (- 1) > 2 (- 1) 4 (x - 1) > - 2 \frac{4 (x - 1)}{4} > \frac{- 2}{4} x - 1 > - \frac{1}{2} x - 1 + 1 > - \frac{1}{2} + 1 x > \frac{1}{2}$

03

Step 3. Draw the solution on number line.

Now, draw the solution $x 鈮� 1$ and $x > \frac{1}{2}$ on the number line:

By combining both the inequalities, we get:

Interval notation: $[1, 鈭�)$

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