Algebra Examples

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

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

Step 1

Multiply each term in the inequality by

12

$12$ .

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

$2$ .

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Step 2.1

Move the leading negative in

- \frac{1}{2}

$- \frac{1}{2}$ into the numerator.

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

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

Step 2.2

Factor

2

$2$ out of

12

$12$ .

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

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

Step 2.4

Rewrite the expression.

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

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

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

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

Step 3

Multiply

- 1

$- 1$ by

6

$6$ .

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

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

Step 4

Cancel the common factor of

12

$12$ .

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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

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

Step 4.2

Rewrite the expression.

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

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

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

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

Step 5

Cancel the common factor of

3

$3$ .

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Step 5.1

Factor

3

$3$ out of

12

$12$ .

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

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

Step 5.2

Cancel the common factor.

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

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

Step 5.3

Rewrite the expression.

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

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

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

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

Step 6

Multiply

2

$2$ by

4

$4$ .

- 6 < 2 x - 13 \leq 8

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

Step 7

Move all terms not containing

x

$x$ from the center section of the inequality.

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Step 7.1

Add

13

$13$ 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

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

Step 7.2

Add

- 6

$- 6$ and

13

$13$ .

7 < 2 x \leq 8 + 13

$7 < 2 x \leq 8 + 13$

Step 7.3

Add

8

$8$ and

13

$13$ .

7 < 2 x \leq 21

$7 < 2 x \leq 21$

7 < 2 x \leq 21

$7 < 2 x \leq 21$

Step 8

Divide each term in the inequality by

2

$2$ .

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

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

Step 9

Cancel the common factor of

2

$2$ .

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Step 9.1

Cancel the common factor.

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

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

Step 9.2

Divide

x

$x$ by

1

$1$ .

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

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

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

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

Step 10

The result can be shown in multiple forms.

Inequality Form:

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

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

Interval Notation:

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

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

Step 11