Question

(\frac{2x - 4}{6} geq -5x + 2)

Explanation:

Step1: Eliminate the denominator

Multiply both sides of the inequality $\frac{2x - 4}{6}\geq - 5x + 2$ by 6 to get rid of the denominator. According to the property of inequalities (multiplying both sides by a positive number does not change the direction of the inequality sign), we have:
$$6\times\frac{2x - 4}{6}\geq6\times(-5x + 2)$$
Simplify both sides:
$$2x-4\geq - 30x + 12$$

Step2: Move the terms with \(x\) to one side and constants to the other side

Add \(30x\) to both sides of the inequality:
$$2x + 30x-4\geq - 30x+30x + 12$$
Simplify:
$$32x-4\geq12$$
Then add 4 to both sides:
$$32x-4 + 4\geq12 + 4$$
Simplify:
$$32x\geq16$$

Step3: Solve for \(x\)

Divide both sides of the inequality $32x\geq16$ by 32. Since 32 is positive, the direction of the inequality sign remains unchanged:
$$x\geq\frac{16}{32}$$
Simplify the fraction:
$$x\geq\frac{1}{2}$$

Answer:

\(x\geq\frac{1}{2}\)