Question: (\[ \frac{2|x - 5|}{7} \geq 10 \] \[ x \geq \boxed{?} \] \[ x \leq \boxed{?} \]\)…

\(\[ \frac{2|x - 5|}{7} \geq 10 \]

\[ x \geq \boxed{?} \]

\[ x \leq \boxed{?} \]\)

Solution

To solve the inequality \[ \frac{2|x - 5|}{7} \geq 10 \] First, multiply both sides by 7 to eliminate the fraction: \[ 2|x - 5| \geq 70 \] Divide both sides by 2: \[ |x - 5| \geq 35 \] This absolute value inequality splits into two separate inequalities: \[ \begin{cases} x - 5 \geq 35 & \\ x - 5 \leq -35 & \end{cases} \] Solve each inequality separately: For \(x - 5 \geq 35\): Add 5 to both sides: \[ x \geq 40 \] For \(x - 5 \leq -35\): Add 5 to both sides: \[ x \leq -30 \] Thus, the solution is: \( x \geq 40 \) or \( x \leq -30 \)