Solve the Inequality for x (x-3)/2-1/6<=(x-5)/3

$\frac{x - 3}{2} - \frac{1}{6} \leq \frac{x - 5}{3}$

Multiply both sides by $3$ .

$(\frac{x - 3}{2} - \frac{1}{6}) \cdot 3 \leq \frac{x - 5}{3} \cdot 3$

Simplify.

Simplify the left side.

Simplify $(\frac{x - 3}{2} - \frac{1}{6}) \cdot 3$ .

To write $\frac{x - 3}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$(\frac{x - 3}{2} \cdot \frac{3}{3} - \frac{1}{6}) \cdot 3 \leq \frac{x - 5}{3} \cdot 3$

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{x - 3}{2}$ by $\frac{3}{3}$ .

$(\frac{(x - 3) \cdot 3}{2 \cdot 3} - \frac{1}{6}) \cdot 3 \leq \frac{x - 5}{3} \cdot 3$

Multiply $2$ by $3$ .

$(\frac{(x - 3) \cdot 3}{6} - \frac{1}{6}) \cdot 3 \leq \frac{x - 5}{3} \cdot 3$

Combine the numerators over the common denominator.

$\frac{(x - 3) \cdot 3 - 1}{6} \cdot 3 \leq \frac{x - 5}{3} \cdot 3$

Simplify the numerator.

Apply the distributive property.

$\frac{x \cdot 3 - 3 \cdot 3 - 1}{6} \cdot 3 \leq \frac{x - 5}{3} \cdot 3$

Move $3$ to the left of $x$ .

$\frac{3 \cdot x - 3 \cdot 3 - 1}{6} \cdot 3 \leq \frac{x - 5}{3} \cdot 3$

Multiply $- 3$ by $3$ .

$\frac{3 \cdot x - 9 - 1}{6} \cdot 3 \leq \frac{x - 5}{3} \cdot 3$

Subtract $1$ from $- 9$ .

$\frac{3 x - 10}{6} \cdot 3 \leq \frac{x - 5}{3} \cdot 3$

Cancel the common factor of $3$ .

Factor $3$ out of $6$ .

$\frac{3 x - 10}{3 (2)} \cdot 3 \leq \frac{x - 5}{3} \cdot 3$

Cancel the common factor.

$\frac{3 x - 10}{3 \cdot 2} \cdot 3 \leq \frac{x - 5}{3} \cdot 3$

Rewrite the expression.

$\frac{3 x - 10}{2} \leq \frac{x - 5}{3} \cdot 3$

Simplify the right side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x - 10}{2} \leq \frac{x - 5}{3} \cdot 3$

Rewrite the expression.

$\frac{3 x - 10}{2} \leq x - 5$

Solve for $x$ .

Multiply both sides by $2$ .

$\frac{3 x - 10}{2} \cdot 2 \leq (x - 5) \cdot 2$

Simplify.

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{3 x - 10}{2} \cdot 2 \leq (x - 5) \cdot 2$

Rewrite the expression.

$3 x - 10 \leq (x - 5) \cdot 2$

Simplify the right side.

Simplify $(x - 5) \cdot 2$ .

Apply the distributive property.

$3 x - 10 \leq x \cdot 2 - 5 \cdot 2$

Simplify the expression.

Move $2$ to the left of $x$ .

$3 x - 10 \leq 2 \cdot x - 5 \cdot 2$

Multiply $- 5$ by $2$ .

$3 x - 10 \leq 2 x - 10$

Solve for $x$ .

Move all terms containing $x$ to the left side of the inequality.

Subtract $2 x$ from both sides of the inequality.

$3 x - 10 - 2 x \leq - 10$

Subtract $2 x$ from $3 x$ .

$x - 10 \leq - 10$

Move all terms not containing $x$ to the right side of the inequality.

Add $10$ to both sides of the inequality.

$x \leq - 10 + 10$

Add $- 10$ and $10$ .

$x \leq 0$

The result can be shown in multiple forms.

Inequality Form:

$x \leq 0$

Interval Notation:

$(- \infty, 0]$

Solve the Inequality for x (x-3)/2-1/6<=(x-5)/3

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