Pre-Algebra Examples

$\frac{1}{4} \cdot (x - 1) < \frac{1}{15} \cdot (4 x - 1)$

Step 1

Simplify $\frac{1}{4} \cdot (x - 1)$ .

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Step 1.1

Rewrite.

$0 + 0 + \frac{1}{4} \cdot (x - 1) < \frac{1}{15} \cdot (4 x - 1)$

Step 1.2

Simplify by adding zeros.

$\frac{1}{4} \cdot (x - 1) < \frac{1}{15} \cdot (4 x - 1)$

Step 1.3

Apply the distributive property.

$\frac{1}{4} x + \frac{1}{4} \cdot - 1 < \frac{1}{15} \cdot (4 x - 1)$

Step 1.4

Combine $\frac{1}{4}$ and $x$ .

$\frac{x}{4} + \frac{1}{4} \cdot - 1 < \frac{1}{15} \cdot (4 x - 1)$

Step 1.5

Combine $\frac{1}{4}$ and $- 1$ .

$\frac{x}{4} + \frac{- 1}{4} < \frac{1}{15} \cdot (4 x - 1)$

Step 1.6

Move the negative in front of the fraction.

$\frac{x}{4} - \frac{1}{4} < \frac{1}{15} \cdot (4 x - 1)$

Step 2

Simplify $\frac{1}{15} \cdot (4 x - 1)$ .

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Step 2.1

Apply the distributive property.

$\frac{x}{4} - \frac{1}{4} < \frac{1}{15} (4 x) + \frac{1}{15} \cdot - 1$

Step 2.2

Multiply $\frac{1}{15} (4 x)$ .

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Step 2.2.1

Combine $4$ and $\frac{1}{15}$ .

$\frac{x}{4} - \frac{1}{4} < \frac{4}{15} x + \frac{1}{15} \cdot - 1$

Step 2.2.2

Combine $\frac{4}{15}$ and $x$ .

$\frac{x}{4} - \frac{1}{4} < \frac{4 x}{15} + \frac{1}{15} \cdot - 1$

Step 2.3

Combine $\frac{1}{15}$ and $- 1$ .

$\frac{x}{4} - \frac{1}{4} < \frac{4 x}{15} + \frac{- 1}{15}$

Step 2.4

Move the negative in front of the fraction.

$\frac{x}{4} - \frac{1}{4} < \frac{4 x}{15} - \frac{1}{15}$

Step 3

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

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Step 3.1

Subtract $\frac{4 x}{15}$ from both sides of the inequality.

$\frac{x}{4} - \frac{1}{4} - \frac{4 x}{15} < - \frac{1}{15}$

Step 3.2

To write $\frac{x}{4}$ as a fraction with a common denominator, multiply by $\frac{15}{15}$ .

$\frac{x}{4} \cdot \frac{15}{15} - \frac{4 x}{15} - \frac{1}{4} < - \frac{1}{15}$

Step 3.3

To write $- \frac{4 x}{15}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{x}{4} \cdot \frac{15}{15} - \frac{4 x}{15} \cdot \frac{4}{4} - \frac{1}{4} < - \frac{1}{15}$

Step 3.4

Write each expression with a common denominator of $60$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.4.1

Multiply $\frac{x}{4}$ by $\frac{15}{15}$ .

$\frac{x \cdot 15}{4 \cdot 15} - \frac{4 x}{15} \cdot \frac{4}{4} - \frac{1}{4} < - \frac{1}{15}$

Step 3.4.2

Multiply $4$ by $15$ .

$\frac{x \cdot 15}{60} - \frac{4 x}{15} \cdot \frac{4}{4} - \frac{1}{4} < - \frac{1}{15}$

Step 3.4.3

Multiply $\frac{4 x}{15}$ by $\frac{4}{4}$ .

$\frac{x \cdot 15}{60} - \frac{4 x \cdot 4}{15 \cdot 4} - \frac{1}{4} < - \frac{1}{15}$

Step 3.4.4

Multiply $15$ by $4$ .

$\frac{x \cdot 15}{60} - \frac{4 x \cdot 4}{60} - \frac{1}{4} < - \frac{1}{15}$

Step 3.5

Combine the numerators over the common denominator.

$\frac{x \cdot 15 - 4 x \cdot 4}{60} - \frac{1}{4} < - \frac{1}{15}$

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 $x \cdot 15 - 4 x \cdot 4$ .

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Step 3.6.1.1.1

Factor $x$ out of $x \cdot 15$ .

$\frac{x (15) - 4 x \cdot 4}{60} - \frac{1}{4} < - \frac{1}{15}$

Step 3.6.1.1.2

Factor $x$ out of $- 4 x \cdot 4$ .

$\frac{x (15) + x (- 4 \cdot 4)}{60} - \frac{1}{4} < - \frac{1}{15}$

Step 3.6.1.1.3

Factor $x$ out of $x (15) + x (- 4 \cdot 4)$ .

$\frac{x (15 - 4 \cdot 4)}{60} - \frac{1}{4} < - \frac{1}{15}$

Step 3.6.1.2

Multiply $- 4$ by $4$ .

$\frac{x (15 - 16)}{60} - \frac{1}{4} < - \frac{1}{15}$

Step 3.6.1.3

Subtract $16$ from $15$ .

$\frac{x \cdot - 1}{60} - \frac{1}{4} < - \frac{1}{15}$

Step 3.6.2

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

$\frac{- 1 \cdot x}{60} - \frac{1}{4} < - \frac{1}{15}$

Step 3.6.3

Move the negative in front of the fraction.

$- \frac{x}{60} - \frac{1}{4} < - \frac{1}{15}$

Step 4

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

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Step 4.1

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

$- \frac{x}{60} < - \frac{1}{15} + \frac{1}{4}$

Step 4.2

To write $- \frac{1}{15}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$- \frac{x}{60} < - \frac{1}{15} \cdot \frac{4}{4} + \frac{1}{4}$

Step 4.3

To write $\frac{1}{4}$ as a fraction with a common denominator, multiply by $\frac{15}{15}$ .

$- \frac{x}{60} < - \frac{1}{15} \cdot \frac{4}{4} + \frac{1}{4} \cdot \frac{15}{15}$

Step 4.4

Write each expression with a common denominator of $60$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.4.1

Multiply $\frac{1}{15}$ by $\frac{4}{4}$ .

$- \frac{x}{60} < - \frac{4}{15 \cdot 4} + \frac{1}{4} \cdot \frac{15}{15}$

Step 4.4.2

Multiply $15$ by $4$ .

$- \frac{x}{60} < - \frac{4}{60} + \frac{1}{4} \cdot \frac{15}{15}$

Step 4.4.3

Multiply $\frac{1}{4}$ by $\frac{15}{15}$ .

$- \frac{x}{60} < - \frac{4}{60} + \frac{15}{4 \cdot 15}$

Step 4.4.4

Multiply $4$ by $15$ .

$- \frac{x}{60} < - \frac{4}{60} + \frac{15}{60}$

Step 4.5

Combine the numerators over the common denominator.

$- \frac{x}{60} < \frac{- 4 + 15}{60}$

Step 4.6

Add $- 4$ and $15$ .

$- \frac{x}{60} < \frac{11}{60}$

Step 5

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

$- x < 11$

Step 6

Divide each term in $- x < 11$ by $- 1$ and simplify.

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Step 6.1

Divide each term in $- x < 11$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} > \frac{11}{- 1}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} > \frac{11}{- 1}$

Step 6.2.2

Divide $x$ by $1$ .

$x > \frac{11}{- 1}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Divide $11$ by $- 1$ .

$x > - 11$

Step 7