Algebra Examples

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

Step 1

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

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Step 1.1

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

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Step 1.1.1

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

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

Step 1.1.2

Subtract $3 x$ from $x$ .

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

Step 1.2

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

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

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

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

Step 1.2.4

Combine the numerators over the common denominator.

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

Step 1.2.5

Simplify the numerator.

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Step 1.2.5.1

Multiply $- 2$ by $2$ .

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

Step 1.2.5.2

Add $- 4$ and $1$ .

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

Step 1.2.6

Move the negative in front of the fraction.

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

Step 1.3

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

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Step 1.3.1

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.

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

Step 1.3.2

Simplify the left side.

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Step 1.3.2.1

Cancel the common factor of $- 2$ .

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Step 1.3.2.1.1

Cancel the common factor.

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

Step 1.3.2.1.2

Divide $x$ by $1$ .

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

Step 1.3.3

Simplify the right side.

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Step 1.3.3.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.3.3.2

Move the negative in front of the fraction.

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

Step 1.3.3.3

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

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Step 1.3.3.3.1

Multiply $- 1$ by $- 1$ .

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

Step 1.3.3.3.2

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

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

Step 1.3.3.3.3

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

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

Step 1.3.3.3.4

Multiply $2$ by $2$ .

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

Step 2

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

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Step 2.1

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

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Step 2.1.1

Subtract $x$ from both sides of the inequality.

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

Step 2.1.2

Subtract $x$ from $3 x$ .

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

Step 2.2

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

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Step 2.2.1

Add $2$ to both sides of the inequality.

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.2.4

Combine the numerators over the common denominator.

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

Step 2.2.5

Simplify the numerator.

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Step 2.2.5.1

Multiply $2$ by $2$ .

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

Step 2.2.5.2

Add $5$ and $4$ .

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $2$ .

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Step 2.3.2.1.1

Cancel the common factor.

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

Step 2.3.2.1.2

Divide $x$ by $1$ .

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

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.3.3.2

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

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Step 2.3.3.2.1

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

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

Step 2.3.3.2.2

Multiply $2$ by $2$ .

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

Step 3

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

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

Step 4

The result can be shown in multiple forms.

Inequality Form:

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

Interval Notation:

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

Step 5