Solve the Inequality for x 1/2x+3<2/3x+5/3

$\frac{1}{2} x + 3 < \frac{2}{3} x + \frac{5}{3}$

Step 1

Combine $\frac{1}{2}$ and $x$ .

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

Step 2

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

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

Step 3

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

Subtract $\frac{2 x}{3}$ from both sides of the inequality.

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

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

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

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

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

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

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

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

Multiply $2$ by $3$ .

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

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

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

Multiply $3$ by $2$ .

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

Combine the numerators over the common denominator.

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

Simplify each term.

Simplify the numerator.

Factor $x$ out of $x \cdot 3 - 2 x \cdot 2$ .

Factor $x$ out of $x \cdot 3$ .

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

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

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

Factor $x$ out of $x (3) + x (- 2 \cdot 2)$ .

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

Multiply $- 2$ by $2$ .

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

Subtract $4$ from $3$ .

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

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

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

Move the negative in front of the fraction.

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

Step 4

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

Subtract $3$ from both sides of the inequality.

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

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

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

Combine $- 3$ and $\frac{3}{3}$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $- 3$ by $3$ .

$- \frac{x}{6} < \frac{5 - 9}{3}$

Subtract $9$ from $5$ .

$- \frac{x}{6} < \frac{- 4}{3}$

Move the negative in front of the fraction.

$- \frac{x}{6} < - \frac{4}{3}$

Step 5

Divide each term in $- \frac{x}{6} < - \frac{4}{3}$ by $- 1$ and simplify.

Divide each term in $- \frac{x}{6} < - \frac{4}{3}$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- \frac{x}{6}}{- 1} > \frac{- \frac{4}{3}}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{\frac{x}{6}}{1} > \frac{- \frac{4}{3}}{- 1}$

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

$\frac{x}{6} > \frac{- \frac{4}{3}}{- 1}$

Simplify the right side.

Dividing two negative values results in a positive value.

$\frac{x}{6} > \frac{\frac{4}{3}}{1}$

Divide $\frac{4}{3}$ by $1$ .

$\frac{x}{6} > \frac{4}{3}$

Step 6

Multiply both sides by $6$ .

$\frac{x}{6} \cdot 6 > \frac{4}{3} \cdot 6$

Step 7

Simplify.

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{x}{6} \cdot 6 > \frac{4}{3} \cdot 6$

Rewrite the expression.

$x > \frac{4}{3} \cdot 6$

Simplify the right side.

Simplify $\frac{4}{3} \cdot 6$ .

Cancel the common factor of $3$ .

Factor $3$ out of $6$ .

$x > \frac{4}{3} \cdot (3 (2))$

Cancel the common factor.

$x > \frac{4}{3} \cdot (3 \cdot 2)$

Rewrite the expression.

$x > 4 \cdot 2$

Multiply $4$ by $2$ .

$x > 8$

Step 8

The result can be shown in multiple forms.

Inequality Form:

$x > 8$

Interval Notation:

$(8, \infty)$

Solve the Inequality for x 1/2x+3<2/3x+5/3

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