Question

StartFraction 5 left parenthesis x minus 6 right parenthesis Over 3 EndFraction
greater than or equalsStartFraction 3 x Over 5 EndFraction
minus15

Answer 1

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:

\[
x \geq -4.6875.
\]