Calculus Examples

$\frac{1}{6} < \frac{2 x - 19}{12} \leq \frac{2}{3}$

Step 1

Multiply each term in the inequality by $12$ .

$\frac{1}{6} \cdot 12 < \frac{2 x - 19}{12} \cdot 12 \leq \frac{2}{3} \cdot 12$

Step 2

Cancel the common factor of $6$ .

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Step 2.1

Factor $6$ out of $12$ .

$\frac{1}{6} \cdot (6 (2)) < \frac{2 x - 19}{12} \cdot 12 \leq \frac{2}{3} \cdot 12$

Step 2.2

Cancel the common factor.

$\frac{1}{6} \cdot (6 \cdot 2) < \frac{2 x - 19}{12} \cdot 12 \leq \frac{2}{3} \cdot 12$

Step 2.3

Rewrite the expression.

$2 < \frac{2 x - 19}{12} \cdot 12 \leq \frac{2}{3} \cdot 12$

Step 3

Cancel the common factor of $12$ .

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Step 3.1

Cancel the common factor.

$2 < \frac{2 x - 19}{12} \cdot 12 \leq \frac{2}{3} \cdot 12$

Step 3.2

Rewrite the expression.

$2 < 2 x - 19 \leq \frac{2}{3} \cdot 12$

Step 4

Cancel the common factor of $3$ .

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Step 4.1

Factor $3$ out of $12$ .

$2 < 2 x - 19 \leq \frac{2}{3} \cdot (3 (4))$

Step 4.2

Cancel the common factor.

$2 < 2 x - 19 \leq \frac{2}{3} \cdot (3 \cdot 4)$

Step 4.3

Rewrite the expression.

$2 < 2 x - 19 \leq 2 \cdot 4$

Step 5

Multiply $2$ by $4$ .

$2 < 2 x - 19 \leq 8$

Step 6

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

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Step 6.1

Add $19$ to each section of the inequality because it does not contain the variable we are trying to solve for.

$2 + 19 < 2 x \leq 8 + 19$

Step 6.2

Add $2$ and $19$ .

$21 < 2 x \leq 8 + 19$

Step 6.3

Add $8$ and $19$ .

$21 < 2 x \leq 27$

Step 7

Divide each term in the inequality by $2$ .

$\frac{21}{2} < \frac{2 x}{2} \leq \frac{27}{2}$

Step 8

Cancel the common factor of $2$ .

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Step 8.1

Cancel the common factor.

$\frac{21}{2} < \frac{2 x}{2} \leq \frac{27}{2}$

Step 8.2

Divide $x$ by $1$ .

$\frac{21}{2} < x \leq \frac{27}{2}$

Step 9

Convert the inequality to interval notation.

$(\frac{21}{2}, \frac{27}{2}]$

Step 10