Algebra Examples

$\frac{1}{2} (\frac{3}{5} x - \frac{1}{2}) > x + \frac{3}{8}$

Step 1

Simplify $\frac{1}{2} \cdot (\frac{3}{5} x - \frac{1}{2})$ .

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Step 1.1

Rewrite.

$0 + 0 + \frac{1}{2} \cdot (\frac{3}{5} x - \frac{1}{2}) > x + \frac{3}{8}$

Step 1.2

Simplify by adding zeros.

$\frac{1}{2} \cdot (\frac{3}{5} x - \frac{1}{2}) > x + \frac{3}{8}$

Step 1.3

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

$\frac{1}{2} \cdot (\frac{3 x}{5} - \frac{1}{2}) > x + \frac{3}{8}$

Step 1.4

Apply the distributive property.

$\frac{1}{2} \cdot \frac{3 x}{5} + \frac{1}{2} (- \frac{1}{2}) > x + \frac{3}{8}$

Step 1.5

Multiply $\frac{1}{2} \cdot \frac{3 x}{5}$ .

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Step 1.5.1

Multiply $\frac{1}{2}$ by $\frac{3 x}{5}$ .

$\frac{3 x}{2 \cdot 5} + \frac{1}{2} (- \frac{1}{2}) > x + \frac{3}{8}$

Step 1.5.2

Multiply $2$ by $5$ .

$\frac{3 x}{10} + \frac{1}{2} (- \frac{1}{2}) > x + \frac{3}{8}$

Step 1.6

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

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Step 1.6.1

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

$\frac{3 x}{10} - \frac{1}{2 \cdot 2} > x + \frac{3}{8}$

Step 1.6.2

Multiply $2$ by $2$ .

$\frac{3 x}{10} - \frac{1}{4} > x + \frac{3}{8}$

Step 2

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

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Step 2.1

Subtract $x$ from both sides of the inequality.

$\frac{3 x}{10} - \frac{1}{4} - x > \frac{3}{8}$

Step 2.2

To write $- x$ as a fraction with a common denominator, multiply by $\frac{10}{10}$ .

$\frac{3 x}{10} - x \cdot \frac{10}{10} - \frac{1}{4} > \frac{3}{8}$

Step 2.3

Combine $- x$ and $\frac{10}{10}$ .

$\frac{3 x}{10} + \frac{- x \cdot 10}{10} - \frac{1}{4} > \frac{3}{8}$

Step 2.4

Combine the numerators over the common denominator.

$\frac{3 x - x \cdot 10}{10} - \frac{1}{4} > \frac{3}{8}$

Step 2.5

Simplify each term.

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Step 2.5.1

Simplify the numerator.

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Step 2.5.1.1

Factor $x$ out of $3 x - x \cdot 10$ .

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Step 2.5.1.1.1

Factor $x$ out of $3 x$ .

$\frac{x \cdot 3 - x \cdot 10}{10} - \frac{1}{4} > \frac{3}{8}$

Step 2.5.1.1.2

Factor $x$ out of $- x \cdot 10$ .

$\frac{x \cdot 3 + x (- 1 \cdot 10)}{10} - \frac{1}{4} > \frac{3}{8}$

Step 2.5.1.1.3

Factor $x$ out of $x \cdot 3 + x (- 1 \cdot 10)$ .

$\frac{x (3 - 1 \cdot 10)}{10} - \frac{1}{4} > \frac{3}{8}$

Step 2.5.1.2

Multiply $- 1$ by $10$ .

$\frac{x (3 - 10)}{10} - \frac{1}{4} > \frac{3}{8}$

Step 2.5.1.3

Subtract $10$ from $3$ .

$\frac{x \cdot - 7}{10} - \frac{1}{4} > \frac{3}{8}$

Step 2.5.2

Move $- 7$ to the left of $x$ .

$\frac{- 7 \cdot x}{10} - \frac{1}{4} > \frac{3}{8}$

Step 2.5.3

Move the negative in front of the fraction.

$- \frac{7 x}{10} - \frac{1}{4} > \frac{3}{8}$

Step 3

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

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Step 3.1

Add $\frac{1}{4}$ to both sides of the inequality.

$- \frac{7 x}{10} > \frac{3}{8} + \frac{1}{4}$

Step 3.2

To write $\frac{1}{4}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{7 x}{10} > \frac{3}{8} + \frac{1}{4} \cdot \frac{2}{2}$

Step 3.3

Write each expression with a common denominator of $8$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.3.1

Multiply $\frac{1}{4}$ by $\frac{2}{2}$ .

$- \frac{7 x}{10} > \frac{3}{8} + \frac{2}{4 \cdot 2}$

Step 3.3.2

Multiply $4$ by $2$ .

$- \frac{7 x}{10} > \frac{3}{8} + \frac{2}{8}$

Step 3.4

Combine the numerators over the common denominator.

$- \frac{7 x}{10} > \frac{3 + 2}{8}$

Step 3.5

Add $3$ and $2$ .

$- \frac{7 x}{10} > \frac{5}{8}$

Step 4

Divide each term in $- \frac{7 x}{10} > \frac{5}{8}$ by $- 1$ and simplify.

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Step 4.1

Divide each term in $- \frac{7 x}{10} > \frac{5}{8}$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- \frac{7 x}{10}}{- 1} < \frac{\frac{5}{8}}{- 1}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Dividing two negative values results in a positive value.

$\frac{\frac{7 x}{10}}{1} < \frac{\frac{5}{8}}{- 1}$

Step 4.2.2

Divide $\frac{7 x}{10}$ by $1$ .

$\frac{7 x}{10} < \frac{\frac{5}{8}}{- 1}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Move the negative one from the denominator of $\frac{\frac{5}{8}}{- 1}$ .

$\frac{7 x}{10} < - 1 \cdot \frac{5}{8}$

Step 4.3.2

Rewrite $- 1 \cdot \frac{5}{8}$ as $- \frac{5}{8}$ .

$\frac{7 x}{10} < - \frac{5}{8}$

Step 5

Multiply both sides by $10$ .

$\frac{7 x}{10} \cdot 10 < - \frac{5}{8} \cdot 10$

Step 6

Simplify.

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Step 6.1

Simplify the left side.

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Step 6.1.1

Cancel the common factor of $10$ .

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Step 6.1.1.1

Cancel the common factor.

$\frac{7 x}{10} \cdot 10 < - \frac{5}{8} \cdot 10$

Step 6.1.1.2

Rewrite the expression.

$7 x < - \frac{5}{8} \cdot 10$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $- \frac{5}{8} \cdot 10$ .

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Step 6.2.1.1

Cancel the common factor of $2$ .

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Step 6.2.1.1.1

Move the leading negative in $- \frac{5}{8}$ into the numerator.

$7 x < \frac{- 5}{8} \cdot 10$

Step 6.2.1.1.2

Factor $2$ out of $8$ .

$7 x < \frac{- 5}{2 (4)} \cdot 10$

Step 6.2.1.1.3

Factor $2$ out of $10$ .

$7 x < \frac{- 5}{2 \cdot 4} \cdot (2 \cdot 5)$

Step 6.2.1.1.4

Cancel the common factor.

$7 x < \frac{- 5}{2 \cdot 4} \cdot (2 \cdot 5)$

Step 6.2.1.1.5

Rewrite the expression.

$7 x < \frac{- 5}{4} \cdot 5$

Step 6.2.1.2

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

$7 x < \frac{- 5 \cdot 5}{4}$

Step 6.2.1.3

Simplify the expression.

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Step 6.2.1.3.1

Multiply $- 5$ by $5$ .

$7 x < \frac{- 25}{4}$

Step 6.2.1.3.2

Move the negative in front of the fraction.

$7 x < - \frac{25}{4}$

Step 7

Divide each term in $7 x < - \frac{25}{4}$ by $7$ and simplify.

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Step 7.1

Divide each term in $7 x < - \frac{25}{4}$ by $7$ .

$\frac{7 x}{7} < \frac{- \frac{25}{4}}{7}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $7$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{7 x}{7} < \frac{- \frac{25}{4}}{7}$

Step 7.2.1.2

Divide $x$ by $1$ .

$x < \frac{- \frac{25}{4}}{7}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Multiply the numerator by the reciprocal of the denominator.

$x < - \frac{25}{4} \cdot \frac{1}{7}$

Step 7.3.2

Multiply $- \frac{25}{4} \cdot \frac{1}{7}$ .

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Step 7.3.2.1

Multiply $\frac{1}{7}$ by $\frac{25}{4}$ .

$x < - \frac{25}{7 \cdot 4}$

Step 7.3.2.2

Multiply $7$ by $4$ .

$x < - \frac{25}{28}$

Step 8

The result can be shown in multiple forms.

Inequality Form:

$x < - \frac{25}{28}$

Interval Notation:

$(- \infty, - \frac{25}{28})$