Algebra Examples

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

Step 1

Multiply each term in the inequality by $5$ .

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

Step 2

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

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Step 2.1

Multiply $5$ by $- 1$ .

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

Step 2.2

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

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

Step 3

Move the negative in front of the fraction.

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

Step 4

Cancel the common factor of $5$ .

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Step 4.1

Cancel the common factor.

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

Step 4.2

Rewrite the expression.

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

Step 5

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

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

Step 6

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

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Step 6.1

Subtract $4$ from each section of the inequality because it does not contain the variable we are trying to solve for.

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

Step 6.2

To write $- 4$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 6.3

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

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

Step 6.4

Combine the numerators over the common denominator.

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

Step 6.5

Simplify the numerator.

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Step 6.5.1

Multiply $- 4$ by $2$ .

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

Step 6.5.2

Subtract $8$ from $- 5$ .

$\frac{- 13}{2} \leq - 3 x \leq \frac{5}{4} - 4$

Step 6.6

Move the negative in front of the fraction.

$- \frac{13}{2} \leq - 3 x \leq \frac{5}{4} - 4$

Step 6.7

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

$- \frac{13}{2} \leq - 3 x \leq \frac{5}{4} - 4 \cdot \frac{4}{4}$

Step 6.8

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

$- \frac{13}{2} \leq - 3 x \leq \frac{5}{4} + \frac{- 4 \cdot 4}{4}$

Step 6.9

Combine the numerators over the common denominator.

$- \frac{13}{2} \leq - 3 x \leq \frac{5 - 4 \cdot 4}{4}$

Step 6.10

Simplify the numerator.

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Step 6.10.1

Multiply $- 4$ by $4$ .

$- \frac{13}{2} \leq - 3 x \leq \frac{5 - 16}{4}$

Step 6.10.2

Subtract $16$ from $5$ .

$- \frac{13}{2} \leq - 3 x \leq \frac{- 11}{4}$

Step 6.11

Move the negative in front of the fraction.

$- \frac{13}{2} \leq - 3 x \leq - \frac{11}{4}$

Step 7

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

$\frac{- \frac{13}{2}}{- 3} \geq \frac{- 3 x}{- 3} \geq \frac{- \frac{11}{4}}{- 3}$

Step 8

Multiply the numerator by the reciprocal of the denominator.

$- \frac{13}{2} \cdot \frac{1}{- 3} \geq \frac{- 3 x}{- 3} \geq \frac{- \frac{11}{4}}{- 3}$

Step 9

Move the negative in front of the fraction.

$- \frac{13}{2} \cdot (- \frac{1}{3}) \geq \frac{- 3 x}{- 3} \geq \frac{- \frac{11}{4}}{- 3}$

Step 10

Multiply $- \frac{13}{2} (- \frac{1}{3})$ .

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Step 10.1

Multiply $- 1$ by $- 1$ .

$1 (\frac{13}{2}) \cdot \frac{1}{3} \geq \frac{- 3 x}{- 3} \geq \frac{- \frac{11}{4}}{- 3}$

Step 10.2

Multiply $\frac{13}{2}$ by $1$ .

$\frac{13}{2} \cdot \frac{1}{3} \geq \frac{- 3 x}{- 3} \geq \frac{- \frac{11}{4}}{- 3}$

Step 10.3

Multiply $\frac{13}{2}$ by $\frac{1}{3}$ .

$\frac{13}{2 \cdot 3} \geq \frac{- 3 x}{- 3} \geq \frac{- \frac{11}{4}}{- 3}$

Step 10.4

Multiply $2$ by $3$ .

$\frac{13}{6} \geq \frac{- 3 x}{- 3} \geq \frac{- \frac{11}{4}}{- 3}$

Step 11

Cancel the common factor of $- 3$ .

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Step 11.1

Cancel the common factor.

$\frac{13}{6} \geq \frac{- 3 x}{- 3} \geq \frac{- \frac{11}{4}}{- 3}$

Step 11.2

Divide $x$ by $1$ .

$\frac{13}{6} \geq x \geq \frac{- \frac{11}{4}}{- 3}$

Step 12

Multiply the numerator by the reciprocal of the denominator.

$\frac{13}{6} \geq x \geq - \frac{11}{4} \cdot \frac{1}{- 3}$

Step 13

Move the negative in front of the fraction.

$\frac{13}{6} \geq x \geq - \frac{11}{4} \cdot (- \frac{1}{3})$

Step 14

Multiply $- \frac{11}{4} (- \frac{1}{3})$ .

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Step 14.1

Multiply $- 1$ by $- 1$ .

$\frac{13}{6} \geq x \geq 1 (\frac{11}{4}) \cdot \frac{1}{3}$

Step 14.2

Multiply $\frac{11}{4}$ by $1$ .

$\frac{13}{6} \geq x \geq \frac{11}{4} \cdot \frac{1}{3}$

Step 14.3

Multiply $\frac{11}{4}$ by $\frac{1}{3}$ .

$\frac{13}{6} \geq x \geq \frac{11}{4 \cdot 3}$

Step 14.4

Multiply $4$ by $3$ .

$\frac{13}{6} \geq x \geq \frac{11}{12}$

Step 15

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

$\frac{11}{12} \leq x \leq \frac{13}{6}$

Step 16

The result can be shown in multiple forms.

Inequality Form:

$\frac{11}{12} \leq x \leq \frac{13}{6}$

Interval Notation:

$[\frac{11}{12}, \frac{13}{6}]$

Step 17