Pre-Algebra Examples

$\frac{1}{5} \cdot (x + 5) < \frac{1}{7} \cdot (3 x + 2)$

Step 1

Simplify $\frac{1}{5} \cdot (x + 5)$ .

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Step 1.1

Rewrite.

$0 + 0 + \frac{1}{5} \cdot (x + 5) < \frac{1}{7} \cdot (3 x + 2)$

Step 1.2

Simplify by adding zeros.

$\frac{1}{5} \cdot (x + 5) < \frac{1}{7} \cdot (3 x + 2)$

Step 1.3

Apply the distributive property.

$\frac{1}{5} x + \frac{1}{5} \cdot 5 < \frac{1}{7} \cdot (3 x + 2)$

Step 1.4

Combine $\frac{1}{5}$ and $x$ .

$\frac{x}{5} + \frac{1}{5} \cdot 5 < \frac{1}{7} \cdot (3 x + 2)$

Step 1.5

Cancel the common factor of $5$ .

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Step 1.5.1

Cancel the common factor.

$\frac{x}{5} + \frac{1}{5} \cdot 5 < \frac{1}{7} \cdot (3 x + 2)$

Step 1.5.2

Rewrite the expression.

$\frac{x}{5} + 1 < \frac{1}{7} \cdot (3 x + 2)$

Step 2

Simplify $\frac{1}{7} \cdot (3 x + 2)$ .

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Step 2.1

Apply the distributive property.

$\frac{x}{5} + 1 < \frac{1}{7} (3 x) + \frac{1}{7} \cdot 2$

Step 2.2

Multiply $\frac{1}{7} (3 x)$ .

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Step 2.2.1

Combine $3$ and $\frac{1}{7}$ .

$\frac{x}{5} + 1 < \frac{3}{7} x + \frac{1}{7} \cdot 2$

Step 2.2.2

Combine $\frac{3}{7}$ and $x$ .

$\frac{x}{5} + 1 < \frac{3 x}{7} + \frac{1}{7} \cdot 2$

Step 2.3

Combine $\frac{1}{7}$ and $2$ .

$\frac{x}{5} + 1 < \frac{3 x}{7} + \frac{2}{7}$

Step 3

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

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Step 3.1

Subtract $\frac{3 x}{7}$ from both sides of the inequality.

$\frac{x}{5} + 1 - \frac{3 x}{7} < \frac{2}{7}$

Step 3.2

To write $\frac{x}{5}$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$\frac{x}{5} \cdot \frac{7}{7} - \frac{3 x}{7} + 1 < \frac{2}{7}$

Step 3.3

To write $- \frac{3 x}{7}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{x}{5} \cdot \frac{7}{7} - \frac{3 x}{7} \cdot \frac{5}{5} + 1 < \frac{2}{7}$

Step 3.4

Write each expression with a common denominator of $35$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.4.1

Multiply $\frac{x}{5}$ by $\frac{7}{7}$ .

$\frac{x \cdot 7}{5 \cdot 7} - \frac{3 x}{7} \cdot \frac{5}{5} + 1 < \frac{2}{7}$

Step 3.4.2

Multiply $5$ by $7$ .

$\frac{x \cdot 7}{35} - \frac{3 x}{7} \cdot \frac{5}{5} + 1 < \frac{2}{7}$

Step 3.4.3

Multiply $\frac{3 x}{7}$ by $\frac{5}{5}$ .

$\frac{x \cdot 7}{35} - \frac{3 x \cdot 5}{7 \cdot 5} + 1 < \frac{2}{7}$

Step 3.4.4

Multiply $7$ by $5$ .

$\frac{x \cdot 7}{35} - \frac{3 x \cdot 5}{35} + 1 < \frac{2}{7}$

Step 3.5

Combine the numerators over the common denominator.

$\frac{x \cdot 7 - 3 x \cdot 5}{35} + 1 < \frac{2}{7}$

Step 3.6

Simplify each term.

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Step 3.6.1

Simplify the numerator.

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Step 3.6.1.1

Factor $x$ out of $x \cdot 7 - 3 x \cdot 5$ .

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Step 3.6.1.1.1

Factor $x$ out of $x \cdot 7$ .

$\frac{x (7) - 3 x \cdot 5}{35} + 1 < \frac{2}{7}$

Step 3.6.1.1.2

Factor $x$ out of $- 3 x \cdot 5$ .

$\frac{x (7) + x (- 3 \cdot 5)}{35} + 1 < \frac{2}{7}$

Step 3.6.1.1.3

Factor $x$ out of $x (7) + x (- 3 \cdot 5)$ .

$\frac{x (7 - 3 \cdot 5)}{35} + 1 < \frac{2}{7}$

Step 3.6.1.2

Multiply $- 3$ by $5$ .

$\frac{x (7 - 15)}{35} + 1 < \frac{2}{7}$

Step 3.6.1.3

Subtract $15$ from $7$ .

$\frac{x \cdot - 8}{35} + 1 < \frac{2}{7}$

Step 3.6.2

Move $- 8$ to the left of $x$ .

$\frac{- 8 \cdot x}{35} + 1 < \frac{2}{7}$

Step 3.6.3

Move the negative in front of the fraction.

$- \frac{8 x}{35} + 1 < \frac{2}{7}$

Step 4

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

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Step 4.1

Subtract $1$ from both sides of the inequality.

$- \frac{8 x}{35} < \frac{2}{7} - 1$

Step 4.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$- \frac{8 x}{35} < \frac{2}{7} - 1 \cdot \frac{7}{7}$

Step 4.3

Combine $- 1$ and $\frac{7}{7}$ .

$- \frac{8 x}{35} < \frac{2}{7} + \frac{- 1 \cdot 7}{7}$

Step 4.4

Combine the numerators over the common denominator.

$- \frac{8 x}{35} < \frac{2 - 1 \cdot 7}{7}$

Step 4.5

Simplify the numerator.

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Step 4.5.1

Multiply $- 1$ by $7$ .

$- \frac{8 x}{35} < \frac{2 - 7}{7}$

Step 4.5.2

Subtract $7$ from $2$ .

$- \frac{8 x}{35} < \frac{- 5}{7}$

Step 4.6

Move the negative in front of the fraction.

$- \frac{8 x}{35} < - \frac{5}{7}$

Step 5

Divide each term in $- \frac{8 x}{35} < - \frac{5}{7}$ by $- 1$ and simplify.

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Step 5.1

Divide each term in $- \frac{8 x}{35} < - \frac{5}{7}$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- \frac{8 x}{35}}{- 1} > \frac{- \frac{5}{7}}{- 1}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Dividing two negative values results in a positive value.

$\frac{\frac{8 x}{35}}{1} > \frac{- \frac{5}{7}}{- 1}$

Step 5.2.2

Divide $\frac{8 x}{35}$ by $1$ .

$\frac{8 x}{35} > \frac{- \frac{5}{7}}{- 1}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Dividing two negative values results in a positive value.

$\frac{8 x}{35} > \frac{\frac{5}{7}}{1}$

Step 5.3.2

Divide $\frac{5}{7}$ by $1$ .

$\frac{8 x}{35} > \frac{5}{7}$

Step 6

Multiply both sides by $35$ .

$\frac{8 x}{35} \cdot 35 > \frac{5}{7} \cdot 35$

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 $35$ .

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Step 7.1.1.1

Cancel the common factor.

$\frac{8 x}{35} \cdot 35 > \frac{5}{7} \cdot 35$

Step 7.1.1.2

Rewrite the expression.

$8 x > \frac{5}{7} \cdot 35$

Step 7.2

Simplify the right side.

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Step 7.2.1

Simplify $\frac{5}{7} \cdot 35$ .

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Step 7.2.1.1

Cancel the common factor of $7$ .

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Step 7.2.1.1.1

Factor $7$ out of $35$ .

$8 x > \frac{5}{7} \cdot (7 (5))$

Step 7.2.1.1.2

Cancel the common factor.

$8 x > \frac{5}{7} \cdot (7 \cdot 5)$

Step 7.2.1.1.3

Rewrite the expression.

$8 x > 5 \cdot 5$

Step 7.2.1.2

Multiply $5$ by $5$ .

$8 x > 25$

Step 8

Divide each term in $8 x > 25$ by $8$ and simplify.

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Step 8.1

Divide each term in $8 x > 25$ by $8$ .

$\frac{8 x}{8} > \frac{25}{8}$

Step 8.2

Simplify the left side.

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Step 8.2.1

Cancel the common factor of $8$ .

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Step 8.2.1.1

Cancel the common factor.

$\frac{8 x}{8} > \frac{25}{8}$

Step 8.2.1.2

Divide $x$ by $1$ .

$x > \frac{25}{8}$

Step 9

The result can be shown in multiple forms.

Inequality Form:

$x > \frac{25}{8}$

Interval Notation:

$(\frac{25}{8}, \infty)$