Algebra Examples

- \frac{1}{2} \leq \frac{7 - 3 x}{5} \leq \frac{1}{4}

$- \frac{1}{2} \leq \frac{7 - 3 x}{5} \leq \frac{1}{4}$

Step 1

Multiply each term in the inequality by

$5$ .

$- \frac{1}{2} \cdot 5 \leq \frac{7 - 3 x}{5} \cdot 5 \leq \frac{1}{4} \cdot 5$

Step 2

Multiply

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

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Step 2.1

Multiply

$5$ by

$- 1$ .

$- 5 (\frac{1}{2}) \leq \frac{7 - 3 x}{5} \cdot 5 \leq \frac{1}{4} \cdot 5$

Step 2.2

Combine

$- 5$ and

$\frac{1}{2}$ .

$\frac{- 5}{2} \leq \frac{7 - 3 x}{5} \cdot 5 \leq \frac{1}{4} \cdot 5$

Step 3

Move the negative in front of the fraction.

$- \frac{5}{2} \leq \frac{7 - 3 x}{5} \cdot 5 \leq \frac{1}{4} \cdot 5$

Step 4

Cancel the common factor of

$5$ .

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Step 4.1

Cancel the common factor.

$- \frac{5}{2} \leq \frac{7 - 3 x}{5} \cdot 5 \leq \frac{1}{4} \cdot 5$

Step 4.2

Rewrite the expression.

$- \frac{5}{2} \leq 7 - 3 x \leq \frac{1}{4} \cdot 5$

Step 5

Combine

$\frac{1}{4}$ and

$5$ .

$- \frac{5}{2} \leq 7 - 3 x \leq \frac{5}{4}$

Step 6

Move all terms not containing

$x$ from the center section of the inequality.

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Step 6.1

Subtract

$7$ from each section of the inequality because it does not contain the variable we are trying to solve for.

$- \frac{5}{2} - 7 \leq - 3 x \leq \frac{5}{4} - 7$

Step 6.2

To write

$- 7$ as a fraction with a common denominator, multiply by

$\frac{2}{2}$ .

$- \frac{5}{2} - 7 \cdot \frac{2}{2} \leq - 3 x \leq \frac{5}{4} - 7$

Step 6.3

Combine

$- 7$ and

$\frac{2}{2}$ .

$- \frac{5}{2} + \frac{- 7 \cdot 2}{2} \leq - 3 x \leq \frac{5}{4} - 7$

Step 6.4

Combine the numerators over the common denominator.

$\frac{- 5 - 7 \cdot 2}{2} \leq - 3 x \leq \frac{5}{4} - 7$

Step 6.5

Simplify the numerator.

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Step 6.5.1

Multiply

$- 7$ by

$2$ .

$\frac{- 5 - 14}{2} \leq - 3 x \leq \frac{5}{4} - 7$

Step 6.5.2

Subtract

$14$ from

$- 5$ .

$\frac{- 19}{2} \leq - 3 x \leq \frac{5}{4} - 7$

Step 6.6

Move the negative in front of the fraction.

$- \frac{19}{2} \leq - 3 x \leq \frac{5}{4} - 7$

Step 6.7

To write

$- 7$ as a fraction with a common denominator, multiply by

$\frac{4}{4}$ .

$- \frac{19}{2} \leq - 3 x \leq \frac{5}{4} - 7 \cdot \frac{4}{4}$

Step 6.8

Combine

$- 7$ and

$\frac{4}{4}$ .

$- \frac{19}{2} \leq - 3 x \leq \frac{5}{4} + \frac{- 7 \cdot 4}{4}$

Step 6.9

Combine the numerators over the common denominator.

$- \frac{19}{2} \leq - 3 x \leq \frac{5 - 7 \cdot 4}{4}$

Step 6.10

Simplify the numerator.

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Step 6.10.1

Multiply

$- 7$ by

$4$ .

$- \frac{19}{2} \leq - 3 x \leq \frac{5 - 28}{4}$

Step 6.10.2

Subtract

$28$ from

$5$ .

$- \frac{19}{2} \leq - 3 x \leq \frac{- 23}{4}$

Step 6.11

Move the negative in front of the fraction.

$- \frac{19}{2} \leq - 3 x \leq - \frac{23}{4}$

Step 7

Divide each term in the inequality by

$- 3$ .

$\frac{- \frac{19}{2}}{- 3} \geq \frac{- 3 x}{- 3} \geq \frac{- \frac{23}{4}}{- 3}$

Step 8

Multiply the numerator by the reciprocal of the denominator.

$- \frac{19}{2} \cdot \frac{1}{- 3} \geq \frac{- 3 x}{- 3} \geq \frac{- \frac{23}{4}}{- 3}$

Step 9

Move the negative in front of the fraction.

$- \frac{19}{2} \cdot (- \frac{1}{3}) \geq \frac{- 3 x}{- 3} \geq \frac{- \frac{23}{4}}{- 3}$

Step 10

Multiply

$- \frac{19}{2} (- \frac{1}{3})$ .

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Step 10.1

Multiply

$- 1$ by

$- 1$ .

$1 (\frac{19}{2}) \cdot \frac{1}{3} \geq \frac{- 3 x}{- 3} \geq \frac{- \frac{23}{4}}{- 3}$

Step 10.2

Multiply

$\frac{19}{2}$ by

$1$ .

$\frac{19}{2} \cdot \frac{1}{3} \geq \frac{- 3 x}{- 3} \geq \frac{- \frac{23}{4}}{- 3}$

Step 10.3

Multiply

$\frac{19}{2}$ by

$\frac{1}{3}$ .

$\frac{19}{2 \cdot 3} \geq \frac{- 3 x}{- 3} \geq \frac{- \frac{23}{4}}{- 3}$

Step 10.4

Multiply

$2$ by

$3$ .

$\frac{19}{6} \geq \frac{- 3 x}{- 3} \geq \frac{- \frac{23}{4}}{- 3}$

Step 11

Cancel the common factor of

$- 3$ .

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Step 11.1

Cancel the common factor.

$\frac{19}{6} \geq \frac{- 3 x}{- 3} \geq \frac{- \frac{23}{4}}{- 3}$

Step 11.2

Divide

$x$ by

$1$ .

$\frac{19}{6} \geq x \geq \frac{- \frac{23}{4}}{- 3}$

Step 12

Multiply the numerator by the reciprocal of the denominator.

$\frac{19}{6} \geq x \geq - \frac{23}{4} \cdot \frac{1}{- 3}$

Step 13

Move the negative in front of the fraction.

$\frac{19}{6} \geq x \geq - \frac{23}{4} \cdot (- \frac{1}{3})$

Step 14

Multiply

$- \frac{23}{4} (- \frac{1}{3})$ .

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Step 14.1

Multiply

$- 1$ by

$- 1$ .

$\frac{19}{6} \geq x \geq 1 (\frac{23}{4}) \cdot \frac{1}{3}$

Step 14.2

Multiply

$\frac{23}{4}$ by

$1$ .

$\frac{19}{6} \geq x \geq \frac{23}{4} \cdot \frac{1}{3}$

Step 14.3

Multiply

$\frac{23}{4}$ by

$\frac{1}{3}$ .

$\frac{19}{6} \geq x \geq \frac{23}{4 \cdot 3}$

Step 14.4

Multiply

$4$ by

$3$ .

$\frac{19}{6} \geq x \geq \frac{23}{12}$

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.

$\frac{23}{12} \leq x \leq \frac{19}{6}$

Step 16

The result can be shown in multiple forms.

Inequality Form:

$\frac{23}{12} \leq x \leq \frac{19}{6}$

Interval Notation:

$[\frac{23}{12}, \frac{19}{6}]$

Step 17