To solve the absolute value inequality [tex]\( \frac{2|x-5|}{7} \geq 10 \)[/tex]:
First, isolate the absolute value expression:
[tex]\[
\frac{2|x-5|}{7} \geq 10
\][/tex]
Multiply both sides of the inequality by 7 to clear the denominator:
[tex]\[
2|x-5| \geq 70
\][/tex]
Next, divide both sides by 2 to further isolate the absolute value:
[tex]\[
|x-5| \geq 35
\][/tex]
The absolute value inequality [tex]\( |x-5| \geq 35 \)[/tex] can be interpreted as two separate inequalities:
[tex]\[
x - 5 \geq 35 \quad \text{or} \quad x - 5 \leq -35
\][/tex]
Solve each inequality separately:
1. [tex]\( x - 5 \geq 35 \)[/tex]:
Add 5 to both sides:
[tex]\[
x \geq 40
\][/tex]
2. [tex]\( x - 5 \leq -35 \)[/tex]:
Add 5 to both sides:
[tex]\[
x \leq -30
\][/tex]
Thus, the solution to the inequality [tex]\( \frac{2|x-5|}{7} \geq 10 \)[/tex] consists of two parts:
[tex]\[
x \geq 40 \quad \text{or} \quad x \leq -30
\][/tex]
To summarize, the positive absolute value for [tex]\( x \)[/tex] is:
[tex]\[
\begin{array}{l}
x \geq 40 \\
x \leq -30
\end{array}
\][/tex]
