Algebra Examples

$- \frac{1}{2} \leq \frac{8 - 3 x}{5} \leq \frac{1}{4}$

Step 1

Multiply each term in the inequality by $5$ .

$- \frac{1}{2} \cdot 5 \leq \frac{8 - 3 x}{5} \cdot 5 \leq \frac{1}{4} \cdot 5$

Step 2

Multiply $- \frac{1}{2} \cdot 5$ .

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Step 2.1

Multiply $5$ by $- 1$ .

$- 5 (\frac{1}{2}) \leq \frac{8 - 3 x}{5} \cdot 5 \leq \frac{1}{4} \cdot 5$

Step 2.2

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

$\frac{- 5}{2} \leq \frac{8 - 3 x}{5} \cdot 5 \leq \frac{1}{4} \cdot 5$

Step 3

Move the negative in front of the fraction.

$- \frac{5}{2} \leq \frac{8 - 3 x}{5} \cdot 5 \leq \frac{1}{4} \cdot 5$

Step 4

Cancel the common factor of $5$ .

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Step 4.1

Cancel the common factor.

$- \frac{5}{2} \leq \frac{8 - 3 x}{5} \cdot 5 \leq \frac{1}{4} \cdot 5$

Step 4.2

Rewrite the expression.

$- \frac{5}{2} \leq 8 - 3 x \leq \frac{1}{4} \cdot 5$

Step 5

Combine $\frac{1}{4}$ and $5$ .

$- \frac{5}{2} \leq 8 - 3 x \leq \frac{5}{4}$

Step 6

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

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Step 6.1

Subtract $8$ from each section of the inequality because it does not contain the variable we are trying to solve for.

$- \frac{5}{2} - 8 \leq - 3 x \leq \frac{5}{4} - 8$

Step 6.2

To write $- 8$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{5}{2} - 8 \cdot \frac{2}{2} \leq - 3 x \leq \frac{5}{4} - 8$

Step 6.3

Combine $- 8$ and $\frac{2}{2}$ .

$- \frac{5}{2} + \frac{- 8 \cdot 2}{2} \leq - 3 x \leq \frac{5}{4} - 8$

Step 6.4

Combine the numerators over the common denominator.

$\frac{- 5 - 8 \cdot 2}{2} \leq - 3 x \leq \frac{5}{4} - 8$

Step 6.5

Simplify the numerator.

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Step 6.5.1

Multiply $- 8$ by $2$ .

$\frac{- 5 - 16}{2} \leq - 3 x \leq \frac{5}{4} - 8$

Step 6.5.2

Subtract $16$ from $- 5$ .

$\frac{- 21}{2} \leq - 3 x \leq \frac{5}{4} - 8$

Step 6.6

Move the negative in front of the fraction.

$- \frac{21}{2} \leq - 3 x \leq \frac{5}{4} - 8$

Step 6.7

To write $- 8$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$- \frac{21}{2} \leq - 3 x \leq \frac{5}{4} - 8 \cdot \frac{4}{4}$

Step 6.8

Combine $- 8$ and $\frac{4}{4}$ .

$- \frac{21}{2} \leq - 3 x \leq \frac{5}{4} + \frac{- 8 \cdot 4}{4}$

Step 6.9

Combine the numerators over the common denominator.

$- \frac{21}{2} \leq - 3 x \leq \frac{5 - 8 \cdot 4}{4}$

Step 6.10

Simplify the numerator.

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Step 6.10.1

Multiply $- 8$ by $4$ .

$- \frac{21}{2} \leq - 3 x \leq \frac{5 - 32}{4}$

Step 6.10.2

Subtract $32$ from $5$ .

$- \frac{21}{2} \leq - 3 x \leq \frac{- 27}{4}$

Step 6.11

Move the negative in front of the fraction.

$- \frac{21}{2} \leq - 3 x \leq - \frac{27}{4}$

Step 7

Divide each term in the inequality by $- 3$ .

$\frac{- \frac{21}{2}}{- 3} \geq \frac{- 3 x}{- 3} \geq \frac{- \frac{27}{4}}{- 3}$

Step 8

Multiply the numerator by the reciprocal of the denominator.

$- \frac{21}{2} \cdot \frac{1}{- 3} \geq \frac{- 3 x}{- 3} \geq \frac{- \frac{27}{4}}{- 3}$

Step 9

Cancel the common factor of $3$ .

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Step 9.1

Move the leading negative in $- \frac{21}{2}$ into the numerator.

$\frac{- 21}{2} \cdot \frac{1}{- 3} \geq \frac{- 3 x}{- 3} \geq \frac{- \frac{27}{4}}{- 3}$

Step 9.2

Factor $3$ out of $- 21$ .

$\frac{3 (- 7)}{2} \cdot \frac{1}{- 3} \geq \frac{- 3 x}{- 3} \geq \frac{- \frac{27}{4}}{- 3}$

Step 9.3

Factor $3$ out of $- 3$ .

$\frac{3 \cdot - 7}{2} \cdot \frac{1}{3 \cdot - 1} \geq \frac{- 3 x}{- 3} \geq \frac{- \frac{27}{4}}{- 3}$

Step 9.4

Cancel the common factor.

$\frac{3 \cdot - 7}{2} \cdot \frac{1}{3 \cdot - 1} \geq \frac{- 3 x}{- 3} \geq \frac{- \frac{27}{4}}{- 3}$

Step 9.5

Rewrite the expression.

$\frac{- 7}{2} \cdot \frac{1}{- 1} \geq \frac{- 3 x}{- 3} \geq \frac{- \frac{27}{4}}{- 3}$

Step 10

Multiply $\frac{- 7}{2}$ by $\frac{1}{- 1}$ .

$\frac{- 7}{2 \cdot - 1} \geq \frac{- 3 x}{- 3} \geq \frac{- \frac{27}{4}}{- 3}$

Step 11

Multiply $2$ by $- 1$ .

$\frac{- 7}{- 2} \geq \frac{- 3 x}{- 3} \geq \frac{- \frac{27}{4}}{- 3}$

Step 12

Dividing two negative values results in a positive value.

$\frac{7}{2} \geq \frac{- 3 x}{- 3} \geq \frac{- \frac{27}{4}}{- 3}$

Step 13

Cancel the common factor of $- 3$ .

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Step 13.1

Cancel the common factor.

$\frac{7}{2} \geq \frac{- 3 x}{- 3} \geq \frac{- \frac{27}{4}}{- 3}$

Step 13.2

Divide $x$ by $1$ .

$\frac{7}{2} \geq x \geq \frac{- \frac{27}{4}}{- 3}$

Step 14

Multiply the numerator by the reciprocal of the denominator.

$\frac{7}{2} \geq x \geq - \frac{27}{4} \cdot \frac{1}{- 3}$

Step 15

Cancel the common factor of $3$ .

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Step 15.1

Move the leading negative in $- \frac{27}{4}$ into the numerator.

$\frac{7}{2} \geq x \geq \frac{- 27}{4} \cdot \frac{1}{- 3}$

Step 15.2

Factor $3$ out of $- 27$ .

$\frac{7}{2} \geq x \geq \frac{3 (- 9)}{4} \cdot \frac{1}{- 3}$

Step 15.3

Factor $3$ out of $- 3$ .

$\frac{7}{2} \geq x \geq \frac{3 \cdot - 9}{4} \cdot \frac{1}{3 \cdot - 1}$

Step 15.4

Cancel the common factor.

$\frac{7}{2} \geq x \geq \frac{3 \cdot - 9}{4} \cdot \frac{1}{3 \cdot - 1}$

Step 15.5

Rewrite the expression.

$\frac{7}{2} \geq x \geq \frac{- 9}{4} \cdot \frac{1}{- 1}$

Step 16

Multiply $\frac{- 9}{4}$ by $\frac{1}{- 1}$ .

$\frac{7}{2} \geq x \geq \frac{- 9}{4 \cdot - 1}$

Step 17

Multiply $4$ by $- 1$ .

$\frac{7}{2} \geq x \geq \frac{- 9}{- 4}$

Step 18

Dividing two negative values results in a positive value.

$\frac{7}{2} \geq x \geq \frac{9}{4}$

Step 19

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

$\frac{9}{4} \leq x \leq \frac{7}{2}$

Step 20

The result can be shown in multiple forms.

Inequality Form:

$\frac{9}{4} \leq x \leq \frac{7}{2}$

Interval Notation:

$[\frac{9}{4}, \frac{7}{2}]$

Step 21