Algebra Examples

$\frac{3}{5} x + \frac{1}{3} < \frac{4}{5} x - \frac{1}{3}$

Step 1

Combine $\frac{3}{5}$ and $x$ .

$\frac{3 x}{5} + \frac{1}{3} < \frac{4}{5} x - \frac{1}{3}$

Step 2

Combine $\frac{4}{5}$ and $x$ .

$\frac{3 x}{5} + \frac{1}{3} < \frac{4 x}{5} - \frac{1}{3}$

Step 3

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

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Step 3.1

Subtract $\frac{4 x}{5}$ from both sides of the inequality.

$\frac{3 x}{5} + \frac{1}{3} - \frac{4 x}{5} < - \frac{1}{3}$

Step 3.2

Combine the numerators over the common denominator.

$\frac{3 x - 4 x}{5} + \frac{1}{3} < - \frac{1}{3}$

Step 3.3

Subtract $4 x$ from $3 x$ .

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

Step 3.4

Move the negative in front of the fraction.

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

Step 4

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

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Step 4.1

Subtract $\frac{1}{3}$ from both sides of the inequality.

$- \frac{x}{5} < - \frac{1}{3} - \frac{1}{3}$

Step 4.2

Combine the numerators over the common denominator.

$- \frac{x}{5} < \frac{- 1 - 1}{3}$

Step 4.3

Subtract $1$ from $- 1$ .

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

Step 4.4

Move the negative in front of the fraction.

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

Step 5

Divide each term in $- \frac{x}{5} < - \frac{2}{3}$ by $- 1$ and simplify.

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Step 5.1

Divide each term in $- \frac{x}{5} < - \frac{2}{3}$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

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

Step 5.2

Simplify the left side.

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Step 5.2.1

Dividing two negative values results in a positive value.

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

Step 5.2.2

Divide $\frac{x}{5}$ by $1$ .

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

Step 5.3

Simplify the right side.

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Step 5.3.1

Dividing two negative values results in a positive value.

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

Step 5.3.2

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

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

Step 6

Multiply both sides by $5$ .

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

Step 7

Simplify.

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Step 7.1

Simplify the left side.

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Step 7.1.1

Cancel the common factor of $5$ .

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Step 7.1.1.1

Cancel the common factor.

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

Step 7.1.1.2

Rewrite the expression.

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

Step 7.2

Simplify the right side.

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Step 7.2.1

Multiply $\frac{2}{3} \cdot 5$ .

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Step 7.2.1.1

Combine $\frac{2}{3}$ and $5$ .

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

Step 7.2.1.2

Multiply $2$ by $5$ .

$x > \frac{10}{3}$

Step 8

The result can be shown in multiple forms.

Inequality Form:

$x > \frac{10}{3}$

Interval Notation:

$(\frac{10}{3}, \infty)$