Algebra Examples

$- 8 < \frac{6 - 9 x}{10} \leq 9$

Step 1

Factor $3$ out of $6 - 9 x$ .

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Step 1.1

Factor $3$ out of $6$ .

$- 8 < \frac{3 (2) - 9 x}{10} \leq 9$

Step 1.2

Factor $3$ out of $- 9 x$ .

$- 8 < \frac{3 (2) + 3 (- 3 x)}{10} \leq 9$

Step 1.3

Factor $3$ out of $3 (2) + 3 (- 3 x)$ .

$- 8 < \frac{3 (2 - 3 x)}{10} \leq 9$

Step 2

Multiply each term in the inequality by $10$ .

$- 8 \cdot 10 < \frac{3 (2 - 3 x)}{10} \cdot 10 \leq 9 \cdot 10$

Step 3

Multiply $- 8$ by $10$ .

$- 80 < \frac{3 (2 - 3 x)}{10} \cdot 10 \leq 9 \cdot 10$

Step 4

Cancel the common factor of $10$ .

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Step 4.1

Cancel the common factor.

$- 80 < \frac{3 (2 - 3 x)}{10} \cdot 10 \leq 9 \cdot 10$

Step 4.2

Rewrite the expression.

$- 80 < 3 (2 - 3 x) \leq 9 \cdot 10$

Step 5

Apply the distributive property.

$- 80 < 3 \cdot 2 + 3 (- 3 x) \leq 9 \cdot 10$

Step 6

Multiply.

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Step 6.1

Multiply $3$ by $2$ .

$- 80 < 6 + 3 (- 3 x) \leq 9 \cdot 10$

Step 6.2

Multiply $- 3$ by $3$ .

$- 80 < 6 - 9 x \leq 9 \cdot 10$

Step 7

Multiply $9$ by $10$ .

$- 80 < 6 - 9 x \leq 90$

Step 8

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

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Step 8.1

Subtract $6$ from each section of the inequality because it does not contain the variable we are trying to solve for.

$- 80 - 6 < - 9 x \leq 90 - 6$

Step 8.2

Subtract $6$ from $- 80$ .

$- 86 < - 9 x \leq 90 - 6$

Step 8.3

Subtract $6$ from $90$ .

$- 86 < - 9 x \leq 84$

Step 9

Divide each term in the inequality by $- 9$ .

$\frac{- 86}{- 9} > \frac{- 9 x}{- 9} \geq \frac{84}{- 9}$

Step 10

Dividing two negative values results in a positive value.

$\frac{86}{9} > \frac{- 9 x}{- 9} \geq \frac{84}{- 9}$

Step 11

Cancel the common factor of $- 9$ .

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Step 11.1

Cancel the common factor.

$\frac{86}{9} > \frac{- 9 x}{- 9} \geq \frac{84}{- 9}$

Step 11.2

Divide $x$ by $1$ .

$\frac{86}{9} > x \geq \frac{84}{- 9}$

Step 12

Cancel the common factor of $84$ and $- 9$ .

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Step 12.1

Factor $3$ out of $84$ .

$\frac{86}{9} > x \geq \frac{3 (28)}{- 9}$

Step 12.2

Cancel the common factors.

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Step 12.2.1

Factor $3$ out of $- 9$ .

$\frac{86}{9} > x \geq \frac{3 \cdot 28}{3 \cdot - 3}$

Step 12.2.2

Cancel the common factor.

$\frac{86}{9} > x \geq \frac{3 \cdot 28}{3 \cdot - 3}$

Step 12.2.3

Rewrite the expression.

$\frac{86}{9} > x \geq \frac{28}{- 3}$

Step 13

Move the negative in front of the fraction.

$\frac{86}{9} > x \geq - \frac{28}{3}$

Step 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.

$- \frac{28}{3} \leq x < \frac{86}{9}$

Step 15

The result can be shown in multiple forms.

Inequality Form:

$- \frac{28}{3} \leq x < \frac{86}{9}$

Interval Notation:

$[- \frac{28}{3}, \frac{86}{9})$

Step 16