Pre-Algebra Examples

$\frac{1}{4} < \frac{2 x - 7}{2} < 5$

Step 1

Multiply each term in the inequality by $2$ .

$\frac{1}{4} \cdot 2 < \frac{2 x - 7}{2} \cdot 2 < 5 \cdot 2$

Step 2

Cancel the common factor of $2$ .

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Step 2.1

Factor $2$ out of $4$ .

$\frac{1}{2 (2)} \cdot 2 < \frac{2 x - 7}{2} \cdot 2 < 5 \cdot 2$

Step 2.2

Cancel the common factor.

$\frac{1}{2 \cdot 2} \cdot 2 < \frac{2 x - 7}{2} \cdot 2 < 5 \cdot 2$

Step 2.3

Rewrite the expression.

$\frac{1}{2} < \frac{2 x - 7}{2} \cdot 2 < 5 \cdot 2$

Step 3

Cancel the common factor of $2$ .

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Step 3.1

Cancel the common factor.

$\frac{1}{2} < \frac{2 x - 7}{2} \cdot 2 < 5 \cdot 2$

Step 3.2

Rewrite the expression.

$\frac{1}{2} < 2 x - 7 < 5 \cdot 2$

Step 4

Multiply $5$ by $2$ .

$\frac{1}{2} < 2 x - 7 < 10$

Step 5

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

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Step 5.1

Add $7$ to each section of the inequality because it does not contain the variable we are trying to solve for.

$\frac{1}{2} + 7 < 2 x < 10 + 7$

Step 5.2

To write $7$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{1}{2} + 7 \cdot \frac{2}{2} < 2 x < 10 + 7$

Step 5.3

Combine $7$ and $\frac{2}{2}$ .

$\frac{1}{2} + \frac{7 \cdot 2}{2} < 2 x < 10 + 7$

Step 5.4

Combine the numerators over the common denominator.

$\frac{1 + 7 \cdot 2}{2} < 2 x < 10 + 7$

Step 5.5

Simplify the numerator.

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Step 5.5.1

Multiply $7$ by $2$ .

$\frac{1 + 14}{2} < 2 x < 10 + 7$

Step 5.5.2

Add $1$ and $14$ .

$\frac{15}{2} < 2 x < 10 + 7$

Step 5.6

Add $10$ and $7$ .

$\frac{15}{2} < 2 x < 17$

Step 6

Divide each term in the inequality by $2$ .

$\frac{\frac{15}{2}}{2} < \frac{2 x}{2} < \frac{17}{2}$

Step 7

Multiply the numerator by the reciprocal of the denominator.

$\frac{15}{2} \cdot \frac{1}{2} < \frac{2 x}{2} < \frac{17}{2}$

Step 8

Multiply $\frac{15}{2} \cdot \frac{1}{2}$ .

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Step 8.1

Multiply $\frac{15}{2}$ by $\frac{1}{2}$ .

$\frac{15}{2 \cdot 2} < \frac{2 x}{2} < \frac{17}{2}$

Step 8.2

Multiply $2$ by $2$ .

$\frac{15}{4} < \frac{2 x}{2} < \frac{17}{2}$

Step 9

Cancel the common factor of $2$ .

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Step 9.1

Cancel the common factor.

$\frac{15}{4} < \frac{2 x}{2} < \frac{17}{2}$

Step 9.2

Divide $x$ by $1$ .

$\frac{15}{4} < x < \frac{17}{2}$

Step 10