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

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

Solve $- \frac{1}{2} + x < 3 x - 2$ for $x$ .

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

Subtract $3 x$ from both sides of the inequality.

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

Subtract $3 x$ from $x$ .

$- \frac{1}{2} - 2 x < - 2$

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

Add $\frac{1}{2}$ to both sides of the inequality.

$- 2 x < - 2 + \frac{1}{2}$

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

$- 2 x < - 2 \cdot \frac{2}{2} + \frac{1}{2}$

Combine $- 2$ and $\frac{2}{2}$ .

$- 2 x < \frac{- 2 \cdot 2}{2} + \frac{1}{2}$

Combine the numerators over the common denominator.

$- 2 x < \frac{- 2 \cdot 2 + 1}{2}$

Simplify the numerator.

Multiply $- 2$ by $2$ .

$- 2 x < \frac{- 4 + 1}{2}$

Add $- 4$ and $1$ .

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

Move the negative in front of the fraction.

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

Divide each term in $- 2 x < - \frac{3}{2}$ by $- 2$ and simplify.

Divide each term in $- 2 x < - \frac{3}{2}$ by $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 2 x}{- 2} > \frac{- \frac{3}{2}}{- 2}$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 x}{- 2} > \frac{- \frac{3}{2}}{- 2}$

Divide $x$ by $1$ .

$x > \frac{- \frac{3}{2}}{- 2}$

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

$x > - \frac{3}{2} \cdot \frac{1}{- 2}$

Move the negative in front of the fraction.

$x > - \frac{3}{2} (- \frac{1}{2})$

Multiply $- \frac{3}{2} (- \frac{1}{2})$ .

Multiply $- 1$ by $- 1$ .

$x > 1 (\frac{3}{2}) \frac{1}{2}$

Multiply $\frac{3}{2}$ by $1$ .

$x > \frac{3}{2} \cdot \frac{1}{2}$

Multiply $\frac{3}{2}$ by $\frac{1}{2}$ .

$x > \frac{3}{2 \cdot 2}$

Multiply $2$ by $2$ .

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

Solve $3 x - 2 \leq \frac{5}{2} + x$ for $x$ .

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

Subtract $x$ from both sides of the inequality.

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

Subtract $x$ from $3 x$ .

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

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

Add $2$ to both sides of the inequality.

$2 x \leq \frac{5}{2} + 2$

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

$2 x \leq \frac{5}{2} + 2 \cdot \frac{2}{2}$

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

$2 x \leq \frac{5}{2} + \frac{2 \cdot 2}{2}$

Combine the numerators over the common denominator.

$2 x \leq \frac{5 + 2 \cdot 2}{2}$

Simplify the numerator.

Multiply $2$ by $2$ .

$2 x \leq \frac{5 + 4}{2}$

Add $5$ and $4$ .

$2 x \leq \frac{9}{2}$

Divide each term in $2 x \leq \frac{9}{2}$ by $2$ and simplify.

Divide each term in $2 x \leq \frac{9}{2}$ by $2$ .

$\frac{2 x}{2} \leq \frac{\frac{9}{2}}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} \leq \frac{\frac{9}{2}}{2}$

Divide $x$ by $1$ .

$x \leq \frac{\frac{9}{2}}{2}$

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

$x \leq \frac{9}{2} \cdot \frac{1}{2}$

Multiply $\frac{9}{2} \cdot \frac{1}{2}$ .

Multiply $\frac{9}{2}$ by $\frac{1}{2}$ .

$x \leq \frac{9}{2 \cdot 2}$

Multiply $2$ by $2$ .

$x \leq \frac{9}{4}$

Find the intersection of $x > \frac{3}{4}$ and $x \leq \frac{9}{4}$ .

$\frac{3}{4} < x \leq \frac{9}{4}$

The result can be shown in multiple forms.

Inequality Form:

$\frac{3}{4} < x \leq \frac{9}{4}$

Interval Notation:

$(\frac{3}{4}, \frac{9}{4}]$

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

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