Algebra Examples

$4 < 2 (1 - 3 x) < 10$

Step 1

Apply the distributive property.

$4 < 2 \cdot 1 + 2 (- 3 x) < 10$

Step 2

Multiply.

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Step 2.1

Multiply $2$ by $1$ .

$4 < 2 + 2 (- 3 x) < 10$

Step 2.2

Multiply $- 3$ by $2$ .

$4 < 2 - 6 x < 10$

Step 3

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

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Step 3.1

Subtract $2$ from each section of the inequality because it does not contain the variable we are trying to solve for.

$4 - 2 < - 6 x < 10 - 2$

Step 3.2

Subtract $2$ from $4$ .

$2 < - 6 x < 10 - 2$

Step 3.3

Subtract $2$ from $10$ .

$2 < - 6 x < 8$

Step 4

Divide each term in the inequality by $- 6$ .

$\frac{2}{- 6} > \frac{- 6 x}{- 6} > \frac{8}{- 6}$

Step 5

Cancel the common factor of $2$ and $- 6$ .

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Step 5.1

Factor $2$ out of $2$ .

$\frac{2 (1)}{- 6} > \frac{- 6 x}{- 6} > \frac{8}{- 6}$

Step 5.2

Cancel the common factors.

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Step 5.2.1

Factor $2$ out of $- 6$ .

$\frac{2 \cdot 1}{2 \cdot - 3} > \frac{- 6 x}{- 6} > \frac{8}{- 6}$

Step 5.2.2

Cancel the common factor.

$\frac{2 \cdot 1}{2 \cdot - 3} > \frac{- 6 x}{- 6} > \frac{8}{- 6}$

Step 5.2.3

Rewrite the expression.

$\frac{1}{- 3} > \frac{- 6 x}{- 6} > \frac{8}{- 6}$

Step 6

Move the negative in front of the fraction.

$- \frac{1}{3} > \frac{- 6 x}{- 6} > \frac{8}{- 6}$

Step 7

Cancel the common factor of $- 6$ .

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Step 7.1

Cancel the common factor.

$- \frac{1}{3} > \frac{- 6 x}{- 6} > \frac{8}{- 6}$

Step 7.2

Divide $x$ by $1$ .

$- \frac{1}{3} > x > \frac{8}{- 6}$

Step 8

Cancel the common factor of $8$ and $- 6$ .

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Step 8.1

Factor $2$ out of $8$ .

$- \frac{1}{3} > x > \frac{2 (4)}{- 6}$

Step 8.2

Cancel the common factors.

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Step 8.2.1

Factor $2$ out of $- 6$ .

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

Step 8.2.2

Cancel the common factor.

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

Step 8.2.3

Rewrite the expression.

$- \frac{1}{3} > x > \frac{4}{- 3}$

Step 9

Move the negative in front of the fraction.

$- \frac{1}{3} > x > - \frac{4}{3}$

Step 10

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

$- \frac{4}{3} < x < - \frac{1}{3}$

Step 11

The result can be shown in multiple forms.

Inequality Form:

$- \frac{4}{3} < x < - \frac{1}{3}$

Interval Notation:

$(- \frac{4}{3}, - \frac{1}{3})$

Step 12