Solve for x 2/3x-2<5/6x+3

$\frac{2}{3} x - 2 < \frac{5}{6} x + 3$

Step 1

Combine $\frac{2}{3}$ and $x$ .

$\frac{2 x}{3} - 2 < \frac{5}{6} x + 3$

Step 2

Combine $\frac{5}{6}$ and $x$ .

$\frac{2 x}{3} - 2 < \frac{5 x}{6} + 3$

Step 3

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

Subtract $\frac{5 x}{6}$ from both sides of the inequality.

$\frac{2 x}{3} - 2 - \frac{5 x}{6} < 3$

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

$\frac{2 x}{3} \cdot \frac{2}{2} - \frac{5 x}{6} - 2 < 3$

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

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

$\frac{2 x \cdot 2}{3 \cdot 2} - \frac{5 x}{6} - 2 < 3$

Multiply $3$ by $2$ .

$\frac{2 x \cdot 2}{6} - \frac{5 x}{6} - 2 < 3$

Combine the numerators over the common denominator.

$\frac{2 x \cdot 2 - 5 x}{6} - 2 < 3$

Simplify each term.

Simplify the numerator.

Factor $x$ out of $2 x \cdot 2 - 5 x$ .

Factor $x$ out of $2 x \cdot 2$ .

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

Factor $x$ out of $- 5 x$ .

$\frac{x (2 \cdot 2) + x \cdot - 5}{6} - 2 < 3$

Factor $x$ out of $x (2 \cdot 2) + x \cdot - 5$ .

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

Multiply $2$ by $2$ .

$\frac{x (4 - 5)}{6} - 2 < 3$

Subtract $5$ from $4$ .

$\frac{x \cdot - 1}{6} - 2 < 3$

Move $- 1$ to the left of $x$ .

$\frac{- 1 \cdot x}{6} - 2 < 3$

Move the negative in front of the fraction.

$- \frac{x}{6} - 2 < 3$

Step 4

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

Add $2$ to both sides of the inequality.

$- \frac{x}{6} < 3 + 2$

Add $3$ and $2$ .

$- \frac{x}{6} < 5$

Step 5

Divide each term in $- \frac{x}{6} < 5$ by $- 1$ and simplify.

Divide each term in $- \frac{x}{6} < 5$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- \frac{x}{6}}{- 1} > \frac{5}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{\frac{x}{6}}{1} > \frac{5}{- 1}$

Divide $\frac{x}{6}$ by $1$ .

$\frac{x}{6} > \frac{5}{- 1}$

Simplify the right side.

Divide $5$ by $- 1$ .

$\frac{x}{6} > - 5$

Step 6

Multiply both sides by $6$ .

$\frac{x}{6} \cdot 6 > - 5 \cdot 6$

Step 7

Simplify.

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{x}{6} \cdot 6 > - 5 \cdot 6$

Rewrite the expression.

$x > - 5 \cdot 6$

Simplify the right side.

Multiply $- 5$ by $6$ .

$x > - 30$

Step 8

The result can be shown in multiple forms.

Inequality Form:

$x > - 30$

Interval Notation:

$(- 30, \infty)$

Solve for x 2/3x-2<5/6x+3

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