Algebra Examples

$\frac{1}{3} \leq \frac{x + 1}{5} < \frac{5}{4}$

Step 1

Multiply each term in the inequality by $5$ .

$\frac{1}{3} \cdot 5 \leq \frac{x + 1}{5} \cdot 5 < \frac{5}{4} \cdot 5$

Step 2

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

$\frac{5}{3} \leq \frac{x + 1}{5} \cdot 5 < \frac{5}{4} \cdot 5$

Step 3

Cancel the common factor of $5$ .

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Step 3.1

Cancel the common factor.

$\frac{5}{3} \leq \frac{x + 1}{5} \cdot 5 < \frac{5}{4} \cdot 5$

Step 3.2

Rewrite the expression.

$\frac{5}{3} \leq x + 1 < \frac{5}{4} \cdot 5$

Step 4

Multiply $\frac{5}{4} \cdot 5$ .

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Step 4.1

Combine $\frac{5}{4}$ and $5$ .

$\frac{5}{3} \leq x + 1 < \frac{5 \cdot 5}{4}$

Step 4.2

Multiply $5$ by $5$ .

$\frac{5}{3} \leq x + 1 < \frac{25}{4}$

Step 5

Move all terms not containing $x$ from the center section of the inequality.

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Step 5.1

Subtract $1$ from each section of the inequality because it does not contain the variable we are trying to solve for.

$\frac{5}{3} - 1 \leq x < \frac{25}{4} - 1$

Step 5.2

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

$\frac{5}{3} - 1 \cdot \frac{3}{3} \leq x < \frac{25}{4} - 1$

Step 5.3

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

$\frac{5}{3} + \frac{- 1 \cdot 3}{3} \leq x < \frac{25}{4} - 1$

Step 5.4

Combine the numerators over the common denominator.

$\frac{5 - 1 \cdot 3}{3} \leq x < \frac{25}{4} - 1$

Step 5.5

Simplify the numerator.

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Step 5.5.1

Multiply $- 1$ by $3$ .

$\frac{5 - 3}{3} \leq x < \frac{25}{4} - 1$

Step 5.5.2

Subtract $3$ from $5$ .

$\frac{2}{3} \leq x < \frac{25}{4} - 1$

Step 5.6

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

$\frac{2}{3} \leq x < \frac{25}{4} - 1 \cdot \frac{4}{4}$

Step 5.7

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

$\frac{2}{3} \leq x < \frac{25}{4} + \frac{- 1 \cdot 4}{4}$

Step 5.8

Combine the numerators over the common denominator.

$\frac{2}{3} \leq x < \frac{25 - 1 \cdot 4}{4}$

Step 5.9

Simplify the numerator.

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Step 5.9.1

Multiply $- 1$ by $4$ .

$\frac{2}{3} \leq x < \frac{25 - 4}{4}$

Step 5.9.2

Subtract $4$ from $25$ .

$\frac{2}{3} \leq x < \frac{21}{4}$

Step 6

The result can be shown in multiple forms.

Inequality Form:

$\frac{2}{3} \leq x < \frac{21}{4}$

Interval Notation:

$[\frac{2}{3}, \frac{21}{4})$

Step 7