Finite Math Examples

$2 < \frac{12 x - 7}{7} < 5$

Step 1

Multiply each term in the inequality by $7$ .

$2 \cdot 7 < \frac{12 x - 7}{7} \cdot 7 < 5 \cdot 7$

Step 2

Multiply $2$ by $7$ .

$14 < \frac{12 x - 7}{7} \cdot 7 < 5 \cdot 7$

Step 3

Cancel the common factor of $7$ .

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Step 3.1

Cancel the common factor.

$14 < \frac{12 x - 7}{7} \cdot 7 < 5 \cdot 7$

Step 3.2

Rewrite the expression.

$14 < 12 x - 7 < 5 \cdot 7$

Step 4

Multiply $5$ by $7$ .

$14 < 12 x - 7 < 35$

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.

$14 + 7 < 12 x < 35 + 7$

Step 5.2

Add $14$ and $7$ .

$21 < 12 x < 35 + 7$

Step 5.3

Add $35$ and $7$ .

$21 < 12 x < 42$

Step 6

Divide each term in the inequality by $12$ .

$\frac{21}{12} < \frac{12 x}{12} < \frac{42}{12}$

Step 7

Cancel the common factor of $21$ and $12$ .

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Step 7.1

Factor $3$ out of $21$ .

$\frac{3 (7)}{12} < \frac{12 x}{12} < \frac{42}{12}$

Step 7.2

Cancel the common factors.

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Step 7.2.1

Factor $3$ out of $12$ .

$\frac{3 \cdot 7}{3 \cdot 4} < \frac{12 x}{12} < \frac{42}{12}$

Step 7.2.2

Cancel the common factor.

$\frac{3 \cdot 7}{3 \cdot 4} < \frac{12 x}{12} < \frac{42}{12}$

Step 7.2.3

Rewrite the expression.

$\frac{7}{4} < \frac{12 x}{12} < \frac{42}{12}$

Step 8

Cancel the common factor of $12$ .

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Step 8.1

Cancel the common factor.

$\frac{7}{4} < \frac{12 x}{12} < \frac{42}{12}$

Step 8.2

Divide $x$ by $1$ .

$\frac{7}{4} < x < \frac{42}{12}$

Step 9

Cancel the common factor of $42$ and $12$ .

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Step 9.1

Factor $6$ out of $42$ .

$\frac{7}{4} < x < \frac{6 (7)}{12}$

Step 9.2

Cancel the common factors.

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Step 9.2.1

Factor $6$ out of $12$ .

$\frac{7}{4} < x < \frac{6 \cdot 7}{6 \cdot 2}$

Step 9.2.2

Cancel the common factor.

$\frac{7}{4} < x < \frac{6 \cdot 7}{6 \cdot 2}$

Step 9.2.3

Rewrite the expression.

$\frac{7}{4} < x < \frac{7}{2}$

Step 10

Convert the inequality to interval notation.

$(\frac{7}{4}, \frac{7}{2})$

Step 11