Pre-Algebra Examples

$\frac{1}{3} \cdot (x - 5) < \frac{1}{8} \cdot (3 x - 1)$

Step 1

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

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Step 1.1

Rewrite.

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

Step 1.2

Simplify by adding zeros.

$\frac{1}{3} \cdot (x - 5) < \frac{1}{8} \cdot (3 x - 1)$

Step 1.3

Apply the distributive property.

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

Step 1.4

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

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

Step 1.5

Combine $\frac{1}{3}$ and $- 5$ .

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

Step 1.6

Move the negative in front of the fraction.

$\frac{x}{3} - \frac{5}{3} < \frac{1}{8} \cdot (3 x - 1)$

Step 2

Simplify $\frac{1}{8} \cdot (3 x - 1)$ .

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Step 2.1

Apply the distributive property.

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

Step 2.2

Multiply $\frac{1}{8} (3 x)$ .

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Step 2.2.1

Combine $3$ and $\frac{1}{8}$ .

$\frac{x}{3} - \frac{5}{3} < \frac{3}{8} x + \frac{1}{8} \cdot - 1$

Step 2.2.2

Combine $\frac{3}{8}$ and $x$ .

$\frac{x}{3} - \frac{5}{3} < \frac{3 x}{8} + \frac{1}{8} \cdot - 1$

Step 2.3

Combine $\frac{1}{8}$ and $- 1$ .

$\frac{x}{3} - \frac{5}{3} < \frac{3 x}{8} + \frac{- 1}{8}$

Step 2.4

Move the negative in front of the fraction.

$\frac{x}{3} - \frac{5}{3} < \frac{3 x}{8} - \frac{1}{8}$

Step 3

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

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Step 3.1

Subtract $\frac{3 x}{8}$ from both sides of the inequality.

$\frac{x}{3} - \frac{5}{3} - \frac{3 x}{8} < - \frac{1}{8}$

Step 3.2

To write $\frac{x}{3}$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$\frac{x}{3} \cdot \frac{8}{8} - \frac{3 x}{8} - \frac{5}{3} < - \frac{1}{8}$

Step 3.3

To write $- \frac{3 x}{8}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{x}{3} \cdot \frac{8}{8} - \frac{3 x}{8} \cdot \frac{3}{3} - \frac{5}{3} < - \frac{1}{8}$

Step 3.4

Write each expression with a common denominator of $24$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.4.1

Multiply $\frac{x}{3}$ by $\frac{8}{8}$ .

$\frac{x \cdot 8}{3 \cdot 8} - \frac{3 x}{8} \cdot \frac{3}{3} - \frac{5}{3} < - \frac{1}{8}$

Step 3.4.2

Multiply $3$ by $8$ .

$\frac{x \cdot 8}{24} - \frac{3 x}{8} \cdot \frac{3}{3} - \frac{5}{3} < - \frac{1}{8}$

Step 3.4.3

Multiply $\frac{3 x}{8}$ by $\frac{3}{3}$ .

$\frac{x \cdot 8}{24} - \frac{3 x \cdot 3}{8 \cdot 3} - \frac{5}{3} < - \frac{1}{8}$

Step 3.4.4

Multiply $8$ by $3$ .

$\frac{x \cdot 8}{24} - \frac{3 x \cdot 3}{24} - \frac{5}{3} < - \frac{1}{8}$

Step 3.5

Combine the numerators over the common denominator.

$\frac{x \cdot 8 - 3 x \cdot 3}{24} - \frac{5}{3} < - \frac{1}{8}$

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 $x \cdot 8 - 3 x \cdot 3$ .

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Step 3.6.1.1.1

Factor $x$ out of $x \cdot 8$ .

$\frac{x (8) - 3 x \cdot 3}{24} - \frac{5}{3} < - \frac{1}{8}$

Step 3.6.1.1.2

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

$\frac{x (8) + x (- 3 \cdot 3)}{24} - \frac{5}{3} < - \frac{1}{8}$

Step 3.6.1.1.3

Factor $x$ out of $x (8) + x (- 3 \cdot 3)$ .

$\frac{x (8 - 3 \cdot 3)}{24} - \frac{5}{3} < - \frac{1}{8}$

Step 3.6.1.2

Multiply $- 3$ by $3$ .

$\frac{x (8 - 9)}{24} - \frac{5}{3} < - \frac{1}{8}$

Step 3.6.1.3

Subtract $9$ from $8$ .

$\frac{x \cdot - 1}{24} - \frac{5}{3} < - \frac{1}{8}$

Step 3.6.2

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

$\frac{- 1 \cdot x}{24} - \frac{5}{3} < - \frac{1}{8}$

Step 3.6.3

Move the negative in front of the fraction.

$- \frac{x}{24} - \frac{5}{3} < - \frac{1}{8}$

Step 4

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

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Step 4.1

Add $\frac{5}{3}$ to both sides of the inequality.

$- \frac{x}{24} < - \frac{1}{8} + \frac{5}{3}$

Step 4.2

To write $- \frac{1}{8}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{x}{24} < - \frac{1}{8} \cdot \frac{3}{3} + \frac{5}{3}$

Step 4.3

To write $\frac{5}{3}$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$- \frac{x}{24} < - \frac{1}{8} \cdot \frac{3}{3} + \frac{5}{3} \cdot \frac{8}{8}$

Step 4.4

Write each expression with a common denominator of $24$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.4.1

Multiply $\frac{1}{8}$ by $\frac{3}{3}$ .

$- \frac{x}{24} < - \frac{3}{8 \cdot 3} + \frac{5}{3} \cdot \frac{8}{8}$

Step 4.4.2

Multiply $8$ by $3$ .

$- \frac{x}{24} < - \frac{3}{24} + \frac{5}{3} \cdot \frac{8}{8}$

Step 4.4.3

Multiply $\frac{5}{3}$ by $\frac{8}{8}$ .

$- \frac{x}{24} < - \frac{3}{24} + \frac{5 \cdot 8}{3 \cdot 8}$

Step 4.4.4

Multiply $3$ by $8$ .

$- \frac{x}{24} < - \frac{3}{24} + \frac{5 \cdot 8}{24}$

Step 4.5

Combine the numerators over the common denominator.

$- \frac{x}{24} < \frac{- 3 + 5 \cdot 8}{24}$

Step 4.6

Simplify the numerator.

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Step 4.6.1

Multiply $5$ by $8$ .

$- \frac{x}{24} < \frac{- 3 + 40}{24}$

Step 4.6.2

Add $- 3$ and $40$ .

$- \frac{x}{24} < \frac{37}{24}$

Step 5

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$- x < 37$

Step 6

Divide each term in $- x < 37$ by $- 1$ and simplify.

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Step 6.1

Divide each term in $- x < 37$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} > \frac{37}{- 1}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} > \frac{37}{- 1}$

Step 6.2.2

Divide $x$ by $1$ .

$x > \frac{37}{- 1}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Divide $37$ by $- 1$ .

$x > - 37$

Step 7

The result can be shown in multiple forms.

Inequality Form:

$x > - 37$

Interval Notation:

$(- 37, \infty)$