To solve the inequality
[tex]\[
\frac{2|x-5|}{7} \geq 10
\][/tex]
we need to isolate the absolute value expression. Let's go through the solution step by step.
1. Eliminate the fraction by multiplying both sides by 7:
[tex]\[
2|x-5| \geq 70
\][/tex]
2. Divide both sides by 2 to isolate the absolute value expression:
[tex]\[
|x-5| \geq 35
\][/tex]
3. Recall the definition of the absolute value inequality:
[tex]\[
|A| \geq B
\][/tex]
This means that:
[tex]\[
A \geq B \quad \text{or} \quad A \leq -B
\][/tex]
In our case, [tex]\( A = x-5 \)[/tex] and [tex]\( B = 35 \)[/tex]. Therefore, we have two inequalities to consider:
[tex]\[
x - 5 \geq 35 \quad \text{or} \quad x - 5 \leq -35
\][/tex]
4. Solve each inequality separately:
- For [tex]\( x - 5 \geq 35 \)[/tex]:
[tex]\[
x \geq 35 + 5
\][/tex]
[tex]\[
x \geq 40
\][/tex]
- For [tex]\( x - 5 \leq -35 \)[/tex]:
[tex]\[
x \leq -35 + 5
\][/tex]
[tex]\[
x \leq -30
\][/tex]
So, the complete solution to the inequality is:
[tex]\[
x \leq -30 \quad \text{or} \quad x \geq 40
\][/tex]
Thus, the positive value of [tex]\( x \)[/tex] that satisfies the inequality is when:
[tex]\[
x \geq 40
\][/tex]
Therefore, we can write:
[tex]\[
\begin{array}{l}
x \geq 40 \\
x \leq -30
\end{array}
\][/tex]
