Algebra Examples

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

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.

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

Step 1.2

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

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

Step 1.3

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

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

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 $\frac{5}{5}$ .

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

Step 1.4.2

Multiply $4$ by $5$ .

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

Step 1.4.3

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

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

Step 1.4.4

Multiply $5$ by $4$ .

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

Step 1.5

Combine the numerators over the common denominator.

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

Step 1.6

Combine the numerators over the common denominator.

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

Step 1.7

Simplify each term.

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Step 1.7.1

Move $5$ to the left of $x$ .

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

Step 1.7.2

Multiply $4$ by $- 1$ .

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

Step 1.8

Subtract $4 x$ from $5 x$ .

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

Step 2

Multiply both sides by $20$ .

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

Step 3

Simplify.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Cancel the common factor of $20$ .

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Step 3.1.1.1

Cancel the common factor.

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

Step 3.1.1.2

Rewrite the expression.

$x - 1 \leq 1 \cdot 20$

Step 3.2

Simplify the right side.

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Step 3.2.1

Multiply $20$ by $1$ .

$x - 1 \leq 20$

Step 4

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

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Step 4.1

Add $1$ to both sides of the inequality.

$x \leq 20 + 1$

Step 4.2

Add $20$ and $1$ .

$x \leq 21$

Step 5

Convert the inequality to interval notation.

$(- \infty, 21]$

Step 6