Algebra Examples

$\frac{2}{3} < \frac{5 - 6 x}{4} < \frac{11}{12}$

Step 1

Multiply each term in the inequality by $4$ .

$\frac{2}{3} \cdot 4 < \frac{5 - 6 x}{4} \cdot 4 < \frac{11}{12} \cdot 4$

Step 2

Multiply $\frac{2}{3} \cdot 4$ .

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Step 2.1

Combine $\frac{2}{3}$ and $4$ .

$\frac{2 \cdot 4}{3} < \frac{5 - 6 x}{4} \cdot 4 < \frac{11}{12} \cdot 4$

Step 2.2

Multiply $2$ by $4$ .

$\frac{8}{3} < \frac{5 - 6 x}{4} \cdot 4 < \frac{11}{12} \cdot 4$

Step 3

Cancel the common factor of $4$ .

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Step 3.1

Cancel the common factor.

$\frac{8}{3} < \frac{5 - 6 x}{4} \cdot 4 < \frac{11}{12} \cdot 4$

Step 3.2

Rewrite the expression.

$\frac{8}{3} < 5 - 6 x < \frac{11}{12} \cdot 4$

Step 4

Cancel the common factor of $4$ .

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Step 4.1

Factor $4$ out of $12$ .

$\frac{8}{3} < 5 - 6 x < \frac{11}{4 (3)} \cdot 4$

Step 4.2

Cancel the common factor.

$\frac{8}{3} < 5 - 6 x < \frac{11}{4 \cdot 3} \cdot 4$

Step 4.3

Rewrite the expression.

$\frac{8}{3} < 5 - 6 x < \frac{11}{3}$

Step 5

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

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Step 5.1

Subtract $5$ from each section of the inequality because it does not contain the variable we are trying to solve for.

$\frac{8}{3} - 5 < - 6 x < \frac{11}{3} - 5$

Step 5.2

To write $- 5$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{8}{3} - 5 \cdot \frac{3}{3} < - 6 x < \frac{11}{3} - 5$

Step 5.3

Combine $- 5$ and $\frac{3}{3}$ .

$\frac{8}{3} + \frac{- 5 \cdot 3}{3} < - 6 x < \frac{11}{3} - 5$

Step 5.4

Combine the numerators over the common denominator.

$\frac{8 - 5 \cdot 3}{3} < - 6 x < \frac{11}{3} - 5$

Step 5.5

Simplify the numerator.

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Step 5.5.1

Multiply $- 5$ by $3$ .

$\frac{8 - 15}{3} < - 6 x < \frac{11}{3} - 5$

Step 5.5.2

Subtract $15$ from $8$ .

$\frac{- 7}{3} < - 6 x < \frac{11}{3} - 5$

Step 5.6

Move the negative in front of the fraction.

$- \frac{7}{3} < - 6 x < \frac{11}{3} - 5$

Step 5.7

To write $- 5$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{7}{3} < - 6 x < \frac{11}{3} - 5 \cdot \frac{3}{3}$

Step 5.8

Combine $- 5$ and $\frac{3}{3}$ .

$- \frac{7}{3} < - 6 x < \frac{11}{3} + \frac{- 5 \cdot 3}{3}$

Step 5.9

Combine the numerators over the common denominator.

$- \frac{7}{3} < - 6 x < \frac{11 - 5 \cdot 3}{3}$

Step 5.10

Simplify the numerator.

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Step 5.10.1

Multiply $- 5$ by $3$ .

$- \frac{7}{3} < - 6 x < \frac{11 - 15}{3}$

Step 5.10.2

Subtract $15$ from $11$ .

$- \frac{7}{3} < - 6 x < \frac{- 4}{3}$

Step 5.11

Move the negative in front of the fraction.

$- \frac{7}{3} < - 6 x < - \frac{4}{3}$

Step 6

Divide each term in the inequality by $- 6$ .

$\frac{- \frac{7}{3}}{- 6} > \frac{- 6 x}{- 6} > \frac{- \frac{4}{3}}{- 6}$

Step 7

Multiply the numerator by the reciprocal of the denominator.

$- \frac{7}{3} \cdot \frac{1}{- 6} > \frac{- 6 x}{- 6} > \frac{- \frac{4}{3}}{- 6}$

Step 8

Move the negative in front of the fraction.

$- \frac{7}{3} \cdot (- \frac{1}{6}) > \frac{- 6 x}{- 6} > \frac{- \frac{4}{3}}{- 6}$

Step 9

Multiply $- \frac{7}{3} (- \frac{1}{6})$ .

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Step 9.1

Multiply $- 1$ by $- 1$ .

$1 (\frac{7}{3}) \cdot \frac{1}{6} > \frac{- 6 x}{- 6} > \frac{- \frac{4}{3}}{- 6}$

Step 9.2

Multiply $\frac{7}{3}$ by $1$ .

$\frac{7}{3} \cdot \frac{1}{6} > \frac{- 6 x}{- 6} > \frac{- \frac{4}{3}}{- 6}$

Step 9.3

Multiply $\frac{7}{3}$ by $\frac{1}{6}$ .

$\frac{7}{3 \cdot 6} > \frac{- 6 x}{- 6} > \frac{- \frac{4}{3}}{- 6}$

Step 9.4

Multiply $3$ by $6$ .

$\frac{7}{18} > \frac{- 6 x}{- 6} > \frac{- \frac{4}{3}}{- 6}$

Step 10

Cancel the common factor of $- 6$ .

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Step 10.1

Cancel the common factor.

$\frac{7}{18} > \frac{- 6 x}{- 6} > \frac{- \frac{4}{3}}{- 6}$

Step 10.2

Divide $x$ by $1$ .

$\frac{7}{18} > x > \frac{- \frac{4}{3}}{- 6}$

Step 11

Multiply the numerator by the reciprocal of the denominator.

$\frac{7}{18} > x > - \frac{4}{3} \cdot \frac{1}{- 6}$

Step 12

Cancel the common factor of $2$ .

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Step 12.1

Move the leading negative in $- \frac{4}{3}$ into the numerator.

$\frac{7}{18} > x > \frac{- 4}{3} \cdot \frac{1}{- 6}$

Step 12.2

Factor $2$ out of $- 4$ .

$\frac{7}{18} > x > \frac{2 (- 2)}{3} \cdot \frac{1}{- 6}$

Step 12.3

Factor $2$ out of $- 6$ .

$\frac{7}{18} > x > \frac{2 \cdot - 2}{3} \cdot \frac{1}{2 \cdot - 3}$

Step 12.4

Cancel the common factor.

$\frac{7}{18} > x > \frac{2 \cdot - 2}{3} \cdot \frac{1}{2 \cdot - 3}$

Step 12.5

Rewrite the expression.

$\frac{7}{18} > x > \frac{- 2}{3} \cdot \frac{1}{- 3}$

Step 13

Multiply $\frac{- 2}{3}$ by $\frac{1}{- 3}$ .

$\frac{7}{18} > x > \frac{- 2}{3 \cdot - 3}$

Step 14

Multiply $3$ by $- 3$ .

$\frac{7}{18} > x > \frac{- 2}{- 9}$

Step 15

Dividing two negative values results in a positive value.

$\frac{7}{18} > x > \frac{2}{9}$

Step 16

Rewrite the interval so that the left value is less than the right value. This is the correct way to write an interval solution.

$\frac{2}{9} < x < \frac{7}{18}$

Step 17

The result can be shown in multiple forms.

Inequality Form:

$\frac{2}{9} < x < \frac{7}{18}$

Interval Notation:

$(\frac{2}{9}, \frac{7}{18})$

Step 18