Algebra Examples

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

Step 1

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

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Step 1.1

Subtract $x$ from both sides of the inequality.

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

Step 1.2

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

$\frac{2 x}{3} - x \cdot \frac{3}{3} + \frac{1}{2} > - 5$

Step 1.3

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

$\frac{2 x}{3} + \frac{- x \cdot 3}{3} + \frac{1}{2} > - 5$

Step 1.4

Combine the numerators over the common denominator.

$\frac{2 x - x \cdot 3}{3} + \frac{1}{2} > - 5$

Step 1.5

Simplify each term.

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Step 1.5.1

Simplify the numerator.

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Step 1.5.1.1

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

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Step 1.5.1.1.1

Factor $x$ out of $2 x$ .

$\frac{x \cdot 2 - x \cdot 3}{3} + \frac{1}{2} > - 5$

Step 1.5.1.1.2

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

$\frac{x \cdot 2 + x (- 1 \cdot 3)}{3} + \frac{1}{2} > - 5$

Step 1.5.1.1.3

Factor $x$ out of $x \cdot 2 + x (- 1 \cdot 3)$ .

$\frac{x (2 - 1 \cdot 3)}{3} + \frac{1}{2} > - 5$

Step 1.5.1.2

Multiply $- 1$ by $3$ .

$\frac{x (2 - 3)}{3} + \frac{1}{2} > - 5$

Step 1.5.1.3

Subtract $3$ from $2$ .

$\frac{x \cdot - 1}{3} + \frac{1}{2} > - 5$

Step 1.5.2

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

$\frac{- 1 \cdot x}{3} + \frac{1}{2} > - 5$

Step 1.5.3

Move the negative in front of the fraction.

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

Step 2

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

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Step 2.1

Subtract $\frac{1}{2}$ from both sides of the inequality.

$- \frac{x}{3} > - 5 - \frac{1}{2}$

Step 2.2

To write $- 5$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{x}{3} > - 5 \cdot \frac{2}{2} - \frac{1}{2}$

Step 2.3

Combine $- 5$ and $\frac{2}{2}$ .

$- \frac{x}{3} > \frac{- 5 \cdot 2}{2} - \frac{1}{2}$

Step 2.4

Combine the numerators over the common denominator.

$- \frac{x}{3} > \frac{- 5 \cdot 2 - 1}{2}$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Multiply $- 5$ by $2$ .

$- \frac{x}{3} > \frac{- 10 - 1}{2}$

Step 2.5.2

Subtract $1$ from $- 10$ .

$- \frac{x}{3} > \frac{- 11}{2}$

Step 2.6

Move the negative in front of the fraction.

$- \frac{x}{3} > - \frac{11}{2}$

Step 3

Divide each term in $- \frac{x}{3} > - \frac{11}{2}$ by $- 1$ and simplify.

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Step 3.1

Divide each term in $- \frac{x}{3} > - \frac{11}{2}$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- \frac{x}{3}}{- 1} < \frac{- \frac{11}{2}}{- 1}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Dividing two negative values results in a positive value.

$\frac{\frac{x}{3}}{1} < \frac{- \frac{11}{2}}{- 1}$

Step 3.2.2

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

$\frac{x}{3} < \frac{- \frac{11}{2}}{- 1}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Dividing two negative values results in a positive value.

$\frac{x}{3} < \frac{\frac{11}{2}}{1}$

Step 3.3.2

Divide $\frac{11}{2}$ by $1$ .

$\frac{x}{3} < \frac{11}{2}$

Step 4

Multiply both sides by $3$ .

$\frac{x}{3} \cdot 3 < \frac{11}{2} \cdot 3$

Step 5

Simplify.

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Step 5.1

Simplify the left side.

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Step 5.1.1

Cancel the common factor of $3$ .

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Step 5.1.1.1

Cancel the common factor.

$\frac{x}{3} \cdot 3 < \frac{11}{2} \cdot 3$

Step 5.1.1.2

Rewrite the expression.

$x < \frac{11}{2} \cdot 3$

Step 5.2

Simplify the right side.

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Step 5.2.1

Multiply $\frac{11}{2} \cdot 3$ .

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Step 5.2.1.1

Combine $\frac{11}{2}$ and $3$ .

$x < \frac{11 \cdot 3}{2}$

Step 5.2.1.2

Multiply $11$ by $3$ .

$x < \frac{33}{2}$

Step 6

Convert the inequality to interval notation.

$(- \infty, \frac{33}{2})$

Step 7