StartFraction 5 left parenthesis x minus 6 right parenthesis Over 3 EndFraction

greater than or equalsStartFraction 3 x Over 5 EndFraction
minus15

1 answer

To solve the inequality

\[ \frac{5(x - 6)}{3} \geq \frac{3x}{5} - 15, \]

we'll start by eliminating the fractions. The least common multiple of the denominators \(3\) and \(5\) is \(15\). We will multiply the entire inequality by \(15\) to clear the fractions:

\[ 15 \cdot \frac{5(x - 6)}{3} \geq 15 \left(\frac{3x}{5} - 15\right). \]

Calculating each side:

Left side: \[ 15 \cdot \frac{5(x - 6)}{3} = 5 \cdot 5(x - 6) = 25(x - 6) = 25x - 150. \]

Right side: \[ 15 \left(\frac{3x}{5} - 15\right) = 3 \cdot 3x - 15 \cdot 15 = 9x - 225. \]

Now we have the inequality:

\[ 25x - 150 \geq 9x - 225. \]

Next, we'll move the terms involving \(x\) to one side and the constant terms to the other side. Subtract \(9x\) from both sides:

\[ 25x - 9x - 150 \geq -225. \] \[ 16x - 150 \geq -225. \]

Now, add \(150\) to both sides:

\[ 16x \geq -225 + 150. \] \[ 16x \geq -75. \]

Now, divide both sides by \(16\):

\[ x \geq \frac{-75}{16}. \]

Thus, the solution to the inequality is:

\[ x \geq -\frac{75}{16}. \]

In decimal form, \(-\frac{75}{16} \approx -4.6875\). Therefore, the final solution can be expressed as:

\[ x \geq -4.6875. \]