Solve the Inequality for x x+2/3<=2x+3/4

$x + \frac{2}{3} \leq 2 x + \frac{3}{4}$

Step 1

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

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

$x + \frac{2}{3} - 2 x \leq \frac{3}{4}$

Subtract $2 x$ from $x$ .

$- x + \frac{2}{3} \leq \frac{3}{4}$

Step 2

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

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

$- x \leq \frac{3}{4} - \frac{2}{3}$

To write $\frac{3}{4}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- x \leq \frac{3}{4} \cdot \frac{3}{3} - \frac{2}{3}$

To write $- \frac{2}{3}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$- x \leq \frac{3}{4} \cdot \frac{3}{3} - \frac{2}{3} \cdot \frac{4}{4}$

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{3}{4}$ by $\frac{3}{3}$ .

$- x \leq \frac{3 \cdot 3}{4 \cdot 3} - \frac{2}{3} \cdot \frac{4}{4}$

Multiply $4$ by $3$ .

$- x \leq \frac{3 \cdot 3}{12} - \frac{2}{3} \cdot \frac{4}{4}$

Multiply $\frac{2}{3}$ by $\frac{4}{4}$ .

$- x \leq \frac{3 \cdot 3}{12} - \frac{2 \cdot 4}{3 \cdot 4}$

Multiply $3$ by $4$ .

$- x \leq \frac{3 \cdot 3}{12} - \frac{2 \cdot 4}{12}$

Combine the numerators over the common denominator.

$- x \leq \frac{3 \cdot 3 - 2 \cdot 4}{12}$

Simplify the numerator.

Multiply $3$ by $3$ .

$- x \leq \frac{9 - 2 \cdot 4}{12}$

Multiply $- 2$ by $4$ .

$- x \leq \frac{9 - 8}{12}$

Subtract $8$ from $9$ .

$- x \leq \frac{1}{12}$

Step 3

Divide each term in $- x \leq \frac{1}{12}$ by $- 1$ and simplify.

Divide each term in $- x \leq \frac{1}{12}$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \geq \frac{\frac{1}{12}}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} \geq \frac{\frac{1}{12}}{- 1}$

Divide $x$ by $1$ .

$x \geq \frac{\frac{1}{12}}{- 1}$

Simplify the right side.

Move the negative one from the denominator of $\frac{\frac{1}{12}}{- 1}$ .

$x \geq - 1 \cdot \frac{1}{12}$

Rewrite $- 1 \cdot \frac{1}{12}$ as $- \frac{1}{12}$ .

$x \geq - \frac{1}{12}$

Step 4

The result can be shown in multiple forms.

Inequality Form:

$x \geq - \frac{1}{12}$

Interval Notation:

$[- \frac{1}{12}, \infty)$

Solve the Inequality for x x+2/3<=2x+3/4

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