Solve for x 1<1-1/3x<3

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

Step 1

Combine $x$ and $\frac{1}{3}$ .

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

Step 2

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

Subtract $1$ from each section of the inequality because it does not contain the variable we are trying to solve for.

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

Subtract $1$ from $1$ .

$0 < - \frac{x}{3} < 3 - 1$

Multiply each term in the inequality by $3$ .

$0 \cdot 3 < - \frac{x}{3} \cdot 3 < 3 \cdot 3 - 1 \cdot 3$

Multiply $0$ by $3$ .

$0 < - \frac{x}{3} \cdot 3 < 3 \cdot 3 - 1 \cdot 3$

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{x}{3}$ into the numerator.

$0 < \frac{- x}{3} \cdot 3 < 3 \cdot 3 - 1 \cdot 3$

Cancel the common factor.

$0 < \frac{- x}{3} \cdot 3 < 3 \cdot 3 - 1 \cdot 3$

Rewrite the expression.

$0 < - x < 3 \cdot 3 - 1 \cdot 3$

Simplify each term.

Multiply $3$ by $3$ .

$0 < - x < 9 - 1 \cdot 3$

Multiply $- 1$ by $3$ .

$0 < - x < 9 - 3$

Subtract $3$ from $9$ .

$0 < - x < 6$

Step 3

Multiply each term in the inequality by $- 1$ .

$\frac{0}{- 1} > \frac{- x}{- 1} > \frac{6}{- 1}$

Step 4

Divide $0$ by $- 1$ .

$0 > \frac{- x}{- 1} > \frac{6}{- 1}$

Step 5

Dividing two negative values results in a positive value.

$0 > \frac{x}{1} > \frac{6}{- 1}$

Step 6

Divide $x$ by $1$ .

$0 > x > \frac{6}{- 1}$

Step 7

Divide $6$ by $- 1$ .

$0 > x > - 6$

Step 8

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

$- 6 < x < 0$

Step 9

The result can be shown in multiple forms.

Inequality Form:

$- 6 < x < 0$

Interval Notation:

$(- 6, 0)$

Step 10

Solve for x 1<1-1/3x<3

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