Algebra Examples

$\frac{1}{5} < \frac{2 x - 4}{3} < 6$

Step 1

Factor $2$ out of $2 x - 4$ .

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Step 1.1

Factor $2$ out of $2 x$ .

$\frac{1}{5} < \frac{2 (x) - 4}{3} < 6$

Step 1.2

Factor $2$ out of $- 4$ .

$\frac{1}{5} < \frac{2 x + 2 \cdot - 2}{3} < 6$

Step 1.3

Factor $2$ out of $2 x + 2 \cdot - 2$ .

$\frac{1}{5} < \frac{2 (x - 2)}{3} < 6$

Step 2

Multiply each term in the inequality by $3$ .

$\frac{1}{5} \cdot 3 < \frac{2 (x - 2)}{3} \cdot 3 < 6 \cdot 3$

Step 3

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

$\frac{3}{5} < \frac{2 (x - 2)}{3} \cdot 3 < 6 \cdot 3$

Step 4

Cancel the common factor of $3$ .

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Step 4.1

Cancel the common factor.

$\frac{3}{5} < \frac{2 (x - 2)}{3} \cdot 3 < 6 \cdot 3$

Step 4.2

Rewrite the expression.

$\frac{3}{5} < 2 (x - 2) < 6 \cdot 3$

Step 5

Apply the distributive property.

$\frac{3}{5} < 2 x + 2 \cdot - 2 < 6 \cdot 3$

Step 6

Multiply $2$ by $- 2$ .

$\frac{3}{5} < 2 x - 4 < 6 \cdot 3$

Step 7

Multiply $6$ by $3$ .

$\frac{3}{5} < 2 x - 4 < 18$

Step 8

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

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Step 8.1

Add $4$ to each section of the inequality because it does not contain the variable we are trying to solve for.

$\frac{3}{5} + 4 < 2 x < 18 + 4$

Step 8.2

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

$\frac{3}{5} + 4 \cdot \frac{5}{5} < 2 x < 18 + 4$

Step 8.3

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

$\frac{3}{5} + \frac{4 \cdot 5}{5} < 2 x < 18 + 4$

Step 8.4

Combine the numerators over the common denominator.

$\frac{3 + 4 \cdot 5}{5} < 2 x < 18 + 4$

Step 8.5

Simplify the numerator.

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Step 8.5.1

Multiply $4$ by $5$ .

$\frac{3 + 20}{5} < 2 x < 18 + 4$

Step 8.5.2

Add $3$ and $20$ .

$\frac{23}{5} < 2 x < 18 + 4$

Step 8.6

Add $18$ and $4$ .

$\frac{23}{5} < 2 x < 22$

Step 9

Divide each term in the inequality by $2$ .

$\frac{\frac{23}{5}}{2} < \frac{2 x}{2} < \frac{22}{2}$

Step 10

Multiply the numerator by the reciprocal of the denominator.

$\frac{23}{5} \cdot \frac{1}{2} < \frac{2 x}{2} < \frac{22}{2}$

Step 11

Multiply $\frac{23}{5} \cdot \frac{1}{2}$ .

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Step 11.1

Multiply $\frac{23}{5}$ by $\frac{1}{2}$ .

$\frac{23}{5 \cdot 2} < \frac{2 x}{2} < \frac{22}{2}$

Step 11.2

Multiply $5$ by $2$ .

$\frac{23}{10} < \frac{2 x}{2} < \frac{22}{2}$

Step 12

Cancel the common factor of $2$ .

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Step 12.1

Cancel the common factor.

$\frac{23}{10} < \frac{2 x}{2} < \frac{22}{2}$

Step 12.2

Divide $x$ by $1$ .

$\frac{23}{10} < x < \frac{22}{2}$

Step 13

Divide $22$ by $2$ .

$\frac{23}{10} < x < 11$

Step 14

The result can be shown in multiple forms.

Inequality Form:

$\frac{23}{10} < x < 11$

Interval Notation:

$(\frac{23}{10}, 11)$

Step 15