Algebra Examples

$\frac{1}{10} < \frac{1 - 2 x}{5} \leq 1$

Step 1

Multiply each term in the inequality by $5$ .

$\frac{1}{10} \cdot 5 < \frac{1 - 2 x}{5} \cdot 5 \leq 1 \cdot 5$

Step 2

Cancel the common factor of $5$ .

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Step 2.1

Factor $5$ out of $10$ .

$\frac{1}{5 (2)} \cdot 5 < \frac{1 - 2 x}{5} \cdot 5 \leq 1 \cdot 5$

Step 2.2

Cancel the common factor.

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

Step 2.3

Rewrite the expression.

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

Step 3

Cancel the common factor of $5$ .

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Step 3.1

Cancel the common factor.

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

Step 3.2

Rewrite the expression.

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

Step 4

Multiply $5$ by $1$ .

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

Step 5

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

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Step 5.1

Subtract $1$ from each section of the inequality because it does not contain the variable we are trying to solve for.

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

Step 5.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 5.3

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

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

Step 5.4

Combine the numerators over the common denominator.

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

Step 5.5

Simplify the numerator.

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Step 5.5.1

Multiply $- 1$ by $2$ .

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

Step 5.5.2

Subtract $2$ from $1$ .

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

Step 5.6

Move the negative in front of the fraction.

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

Step 5.7

Subtract $1$ from $5$ .

$- \frac{1}{2} < - 2 x \leq 4$

Step 6

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

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

Step 7

Multiply the numerator by the reciprocal of the denominator.

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

Step 8

Move the negative in front of the fraction.

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

Step 9

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

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Step 9.1

Multiply $- 1$ by $- 1$ .

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

Step 9.2

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

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

Step 9.3

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

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

Step 9.4

Multiply $2$ by $2$ .

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

Step 10

Cancel the common factor of $- 2$ .

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Step 10.1

Cancel the common factor.

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

Step 10.2

Divide $x$ by $1$ .

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

Step 11

Divide $4$ by $- 2$ .

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

Step 12

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

$- 2 \leq x < \frac{1}{4}$

Step 13

The result can be shown in multiple forms.

Inequality Form:

$- 2 \leq x < \frac{1}{4}$

Interval Notation:

$[- 2, \frac{1}{4})$

Step 14