Question: The image contains the following mathematical expressions: \[ \frac{|2x + 1|}{3} \geq 5 \]…

The image contains the following mathematical expressions:

\[ \frac{|2x + 1|}{3} \geq 5 \]

Below this inequality, there are two lines with placeholders for solutions:

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

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

Solution

Let’s solve the inequality step by step. Given the inequality: \[ \frac{|2x + 1|}{3} \geq 5 \] First, multiply both sides by 3 to eliminate the fraction: \[ |2x + 1| \geq 15 \] The absolute value inequality \(|2x + 1| \geq 15\) can be split into two separate inequalities: \[ \begin{cases} 2x + 1 \geq 15 & \\ 2x + 1 \leq -15 & \end{cases} \] Solving the first inequality: \[ 2x + 1 \geq 15 \] Subtract 1 from both sides: \[ 2x \geq 14 \] Divide both sides by 2: \[ x \geq 7 \] Solving the second inequality: \[ 2x + 1 \leq -15 \] Subtract 1 from both sides: \[ 2x \leq -16 \] Divide both sides by 2: \[ x \leq -8 \] The solution for \(x\) is: \[ x \geq 7 \quad \text{or} \quad x \leq -8 \]