Algebra Examples

$\frac{x}{4} - \frac{1}{4} \leq \frac{x}{5} + 1$

Step 1

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

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Step 1.1

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

$\frac{x}{4} - \frac{1}{4} - \frac{x}{5} \leq 1$

Step 1.2

To write $\frac{x}{4}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{x}{4} \cdot \frac{5}{5} - \frac{x}{5} - \frac{1}{4} \leq 1$

Step 1.3

To write $- \frac{x}{5}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{x}{4} \cdot \frac{5}{5} - \frac{x}{5} \cdot \frac{4}{4} - \frac{1}{4} \leq 1$

Step 1.4

Write each expression with a common denominator of $20$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.4.1

Multiply $\frac{x}{4}$ by $\frac{5}{5}$ .

$\frac{x \cdot 5}{4 \cdot 5} - \frac{x}{5} \cdot \frac{4}{4} - \frac{1}{4} \leq 1$

Step 1.4.2

Multiply $4$ by $5$ .

$\frac{x \cdot 5}{20} - \frac{x}{5} \cdot \frac{4}{4} - \frac{1}{4} \leq 1$

Step 1.4.3

Multiply $\frac{x}{5}$ by $\frac{4}{4}$ .

$\frac{x \cdot 5}{20} - \frac{x \cdot 4}{5 \cdot 4} - \frac{1}{4} \leq 1$

Step 1.4.4

Multiply $5$ by $4$ .

$\frac{x \cdot 5}{20} - \frac{x \cdot 4}{20} - \frac{1}{4} \leq 1$

Step 1.5

Combine the numerators over the common denominator.

$\frac{x \cdot 5 - x \cdot 4}{20} - \frac{1}{4} \leq 1$

Step 1.6

Simplify each term.

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Step 1.6.1

Simplify the numerator.

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Step 1.6.1.1

Factor $x$ out of $x \cdot 5 - x \cdot 4$ .

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Step 1.6.1.1.1

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

$\frac{x (5) - x \cdot 4}{20} - \frac{1}{4} \leq 1$

Step 1.6.1.1.2

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

$\frac{x (5) + x (- 1 \cdot 4)}{20} - \frac{1}{4} \leq 1$

Step 1.6.1.1.3

Factor $x$ out of $x (5) + x (- 1 \cdot 4)$ .

$\frac{x (5 - 1 \cdot 4)}{20} - \frac{1}{4} \leq 1$

Step 1.6.1.2

Multiply $- 1$ by $4$ .

$\frac{x (5 - 4)}{20} - \frac{1}{4} \leq 1$

Step 1.6.1.3

Subtract $4$ from $5$ .

$\frac{x \cdot 1}{20} - \frac{1}{4} \leq 1$

Step 1.6.2

Multiply $x$ by $1$ .

$\frac{x}{20} - \frac{1}{4} \leq 1$

Step 2

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

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Step 2.1

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

$\frac{x}{20} \leq 1 + \frac{1}{4}$

Step 2.2

Write $1$ as a fraction with a common denominator.

$\frac{x}{20} \leq \frac{4}{4} + \frac{1}{4}$

Step 2.3

Combine the numerators over the common denominator.

$\frac{x}{20} \leq \frac{4 + 1}{4}$

Step 2.4

Add $4$ and $1$ .

$\frac{x}{20} \leq \frac{5}{4}$

Step 3

Multiply both sides by $20$ .

$\frac{x}{20} \cdot 20 \leq \frac{5}{4} \cdot 20$

Step 4

Simplify.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Cancel the common factor of $20$ .

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Step 4.1.1.1

Cancel the common factor.

$\frac{x}{20} \cdot 20 \leq \frac{5}{4} \cdot 20$

Step 4.1.1.2

Rewrite the expression.

$x \leq \frac{5}{4} \cdot 20$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $\frac{5}{4} \cdot 20$ .

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Step 4.2.1.1

Cancel the common factor of $4$ .

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Step 4.2.1.1.1

Factor $4$ out of $20$ .

$x \leq \frac{5}{4} \cdot (4 (5))$

Step 4.2.1.1.2

Cancel the common factor.

$x \leq \frac{5}{4} \cdot (4 \cdot 5)$

Step 4.2.1.1.3

Rewrite the expression.

$x \leq 5 \cdot 5$

Step 4.2.1.2

Multiply $5$ by $5$ .

$x \leq 25$

Step 5

Convert the inequality to interval notation.

$(- \infty, 25]$

Step 6