Pre-Algebra Examples

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

Step 1

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

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

Step 2

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

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Step 2.1

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

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

Step 2.2

Subtract $1$ from $0$ .

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

Step 2.3

Multiply each term in the inequality by $3$ .

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

Step 2.4

Multiply $- 1$ by $3$ .

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

Step 2.5

Cancel the common factor of $3$ .

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Step 2.5.1

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

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

Step 2.5.2

Cancel the common factor.

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

Step 2.5.3

Rewrite the expression.

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

Step 2.6

Simplify each term.

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Step 2.6.1

Multiply $3$ by $1$ .

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

Step 2.6.2

Multiply $- 1$ by $3$ .

$- 3 < - x < 3 - 3$

Step 2.7

Subtract $3$ from $3$ .

$- 3 < - x < 0$

Step 3

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

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

Step 4

Divide $- 3$ by $- 1$ .

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

Step 5

Dividing two negative values results in a positive value.

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

Step 6

Divide $x$ by $1$ .

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

Step 7

Divide $0$ by $- 1$ .

$3 > x > 0$

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.

$0 < x < 3$

Step 9