Algebra Examples

$\frac{1}{2} < \frac{3 - 2 x}{5} < \frac{2}{3}$

Step 1

Solve $\frac{1}{4} > x > - \frac{1}{6}$ .

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Step 1.1

Multiply each term in the inequality by $5$ .

$\frac{1}{2} \cdot 5 < \frac{3 - 2 x}{5} \cdot 5 < \frac{2}{3} \cdot 5$

Step 1.2

Combine $\frac{1}{2}$ and $5$ .

$\frac{5}{2} < \frac{3 - 2 x}{5} \cdot 5 < \frac{2}{3} \cdot 5$

Step 1.3

Cancel the common factor of $5$ .

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Step 1.3.1

Cancel the common factor.

$\frac{5}{2} < \frac{3 - 2 x}{5} \cdot 5 < \frac{2}{3} \cdot 5$

Step 1.3.2

Rewrite the expression.

$\frac{5}{2} < 3 - 2 x < \frac{2}{3} \cdot 5$

Step 1.4

Multiply $\frac{2}{3} \cdot 5$ .

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Step 1.4.1

Combine $\frac{2}{3}$ and $5$ .

$\frac{5}{2} < 3 - 2 x < \frac{2 \cdot 5}{3}$

Step 1.4.2

Multiply $2$ by $5$ .

$\frac{5}{2} < 3 - 2 x < \frac{10}{3}$

Step 1.5

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

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Step 1.5.1

Subtract $3$ from each section of the inequality because it does not contain the variable we are trying to solve for.

$\frac{5}{2} - 3 < - 2 x < \frac{10}{3} - 3$

Step 1.5.2

To write $- 3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{5}{2} - 3 \cdot \frac{2}{2} < - 2 x < \frac{10}{3} - 3$

Step 1.5.3

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

$\frac{5}{2} + \frac{- 3 \cdot 2}{2} < - 2 x < \frac{10}{3} - 3$

Step 1.5.4

Combine the numerators over the common denominator.

$\frac{5 - 3 \cdot 2}{2} < - 2 x < \frac{10}{3} - 3$

Step 1.5.5

Simplify the numerator.

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Step 1.5.5.1

Multiply $- 3$ by $2$ .

$\frac{5 - 6}{2} < - 2 x < \frac{10}{3} - 3$

Step 1.5.5.2

Subtract $6$ from $5$ .

$\frac{- 1}{2} < - 2 x < \frac{10}{3} - 3$

Step 1.5.6

Move the negative in front of the fraction.

$- \frac{1}{2} < - 2 x < \frac{10}{3} - 3$

Step 1.5.7

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

$- \frac{1}{2} < - 2 x < \frac{10}{3} - 3 \cdot \frac{3}{3}$

Step 1.5.8

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

$- \frac{1}{2} < - 2 x < \frac{10}{3} + \frac{- 3 \cdot 3}{3}$

Step 1.5.9

Combine the numerators over the common denominator.

$- \frac{1}{2} < - 2 x < \frac{10 - 3 \cdot 3}{3}$

Step 1.5.10

Simplify the numerator.

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Step 1.5.10.1

Multiply $- 3$ by $3$ .

$- \frac{1}{2} < - 2 x < \frac{10 - 9}{3}$

Step 1.5.10.2

Subtract $9$ from $10$ .

$- \frac{1}{2} < - 2 x < \frac{1}{3}$

Step 1.6

Divide each term in the inequality by $- 2$ .

$\frac{- \frac{1}{2}}{- 2} > \frac{- 2 x}{- 2} > \frac{\frac{1}{3}}{- 2}$

Step 1.7

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.8

Move the negative in front of the fraction.

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

Step 1.9

Multiply $- \frac{1}{2} (- \frac{1}{2})$ .

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Step 1.9.1

Multiply $- 1$ by $- 1$ .

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

Step 1.9.2

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

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

Step 1.9.3

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

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

Step 1.9.4

Multiply $2$ by $2$ .

$\frac{1}{4} > \frac{- 2 x}{- 2} > \frac{\frac{1}{3}}{- 2}$

Step 1.10

Cancel the common factor of $- 2$ .

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Step 1.10.1

Cancel the common factor.

$\frac{1}{4} > \frac{- 2 x}{- 2} > \frac{\frac{1}{3}}{- 2}$

Step 1.10.2

Divide $x$ by $1$ .

$\frac{1}{4} > x > \frac{\frac{1}{3}}{- 2}$

Step 1.11

Multiply the numerator by the reciprocal of the denominator.

$\frac{1}{4} > x > \frac{1}{3} \cdot \frac{1}{- 2}$

Step 1.12

Move the negative in front of the fraction.

$\frac{1}{4} > x > \frac{1}{3} \cdot (- \frac{1}{2})$

Step 1.13

Multiply $\frac{1}{3} (- \frac{1}{2})$ .

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Step 1.13.1

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

$\frac{1}{4} > x > - \frac{1}{3 \cdot 2}$

Step 1.13.2

Multiply $3$ by $2$ .

$\frac{1}{4} > x > - \frac{1}{6}$

Step 1.14

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

$- \frac{1}{6} < x < \frac{1}{4}$

Step 2

Use the inequality $- \frac{1}{6} < x < \frac{1}{4}$ to build the set notation.

${x | - \frac{1}{6} < x < \frac{1}{4}}$

Step 3