To solve the inequality [tex]\(-2(8x - 4) < 2x + 0.5\)[/tex], let's work through it step-by-step:
1. Distribute [tex]\(-2\)[/tex] to the terms inside the parentheses:
[tex]\[
-2(8x - 4) = -2 \cdot 8x + (-2) \cdot (-4) = -16x + 8
\][/tex]
So, the inequality becomes:
[tex]\[
-16x + 8 < 2x + 0.5
\][/tex]
2. Rearrange the inequality to isolate the terms involving [tex]\( x \)[/tex] on one side:
Subtract [tex]\( 2x \)[/tex] from both sides:
[tex]\[
-16x - 2x + 8 < 0.5
\][/tex]
Simplify:
[tex]\[
-18x + 8 < 0.5
\][/tex]
3. Subtract 8 from both sides:
[tex]\[
-18x + 8 - 8 < 0.5 - 8
\][/tex]
Simplify:
[tex]\[
-18x < -7.5
\][/tex]
4. Divide both sides by [tex]\(-18\)[/tex]:
Note that when dividing by a negative number, the inequality sign reverses:
[tex]\[
x > \frac{-7.5}{-18}
\][/tex]
Simplify the fraction:
[tex]\[
x > \frac{7.5}{18}
\][/tex]
Simplify further:
[tex]\[
x > \frac{5}{12}
\][/tex]
Therefore, the solution to the inequality [tex]\(-2(8x - 4) < 2x + 0.5\)[/tex] is:
[tex]\[
x > \frac{5}{12}
\][/tex]
Let's check the provided options:
- [tex]\(x > \frac{1}{6}\)[/tex]
- [tex]\(x < \frac{1}{6}\)[/tex]
- [tex]\(x > 6\)[/tex]
- [tex]\(x < 6\)[/tex]
The correct option is:
[tex]\[
x > \frac{1}{6}
\][/tex]
However, based on correct calculations and the resultant fraction, the actual direct calculation shows:
[tex]\[
x > \frac{5}{12}
\][/tex]
