Algebra Examples

$- \frac{1}{2} < \frac{2 x - 13}{12} \leq \frac{2}{3}$

Step 1

Multiply each term in the inequality by $12$ .

$- \frac{1}{2} \cdot 12 < \frac{2 x - 13}{12} \cdot 12 \leq \frac{2}{3} \cdot 12$

Step 2

Cancel the common factor of $2$ .

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Step 2.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$\frac{- 1}{2} \cdot 12 < \frac{2 x - 13}{12} \cdot 12 \leq \frac{2}{3} \cdot 12$

Step 2.2

Factor $2$ out of $12$ .

$\frac{- 1}{2} \cdot (2 (6)) < \frac{2 x - 13}{12} \cdot 12 \leq \frac{2}{3} \cdot 12$

Step 2.3

Cancel the common factor.

$\frac{- 1}{2} \cdot (2 \cdot 6) < \frac{2 x - 13}{12} \cdot 12 \leq \frac{2}{3} \cdot 12$

Step 2.4

Rewrite the expression.

$- 1 \cdot 6 < \frac{2 x - 13}{12} \cdot 12 \leq \frac{2}{3} \cdot 12$

Step 3

Multiply $- 1$ by $6$ .

$- 6 < \frac{2 x - 13}{12} \cdot 12 \leq \frac{2}{3} \cdot 12$

Step 4

Cancel the common factor of $12$ .

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Step 4.1

Cancel the common factor.

$- 6 < \frac{2 x - 13}{12} \cdot 12 \leq \frac{2}{3} \cdot 12$

Step 4.2

Rewrite the expression.

$- 6 < 2 x - 13 \leq \frac{2}{3} \cdot 12$

Step 5

Cancel the common factor of $3$ .

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Step 5.1

Factor $3$ out of $12$ .

$- 6 < 2 x - 13 \leq \frac{2}{3} \cdot (3 (4))$

Step 5.2

Cancel the common factor.

$- 6 < 2 x - 13 \leq \frac{2}{3} \cdot (3 \cdot 4)$

Step 5.3

Rewrite the expression.

$- 6 < 2 x - 13 \leq 2 \cdot 4$

Step 6

Multiply $2$ by $4$ .

$- 6 < 2 x - 13 \leq 8$

Step 7

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

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Step 7.1

Add $13$ to each section of the inequality because it does not contain the variable we are trying to solve for.

$- 6 + 13 < 2 x \leq 8 + 13$

Step 7.2

Add $- 6$ and $13$ .

$7 < 2 x \leq 8 + 13$

Step 7.3

Add $8$ and $13$ .

$7 < 2 x \leq 21$

Step 8

Divide each term in the inequality by $2$ .

$\frac{7}{2} < \frac{2 x}{2} \leq \frac{21}{2}$

Step 9

Cancel the common factor of $2$ .

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Step 9.1

Cancel the common factor.

$\frac{7}{2} < \frac{2 x}{2} \leq \frac{21}{2}$

Step 9.2

Divide $x$ by $1$ .

$\frac{7}{2} < x \leq \frac{21}{2}$

Step 10

The result can be shown in multiple forms.

Inequality Form:

$\frac{7}{2} < x \leq \frac{21}{2}$

Interval Notation:

$(\frac{7}{2}, \frac{21}{2}]$

Step 11