Solve for x 1<1-1/2x<2

$1 < 1 - \frac{1}{2} x < 2$

Step 1

Combine $x$ and $\frac{1}{2}$ .

$1 < 1 - \frac{x}{2} < 2$

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.

$1 - 1 < - \frac{x}{2} < 2 - 1$

Subtract $1$ from $1$ .

$0 < - \frac{x}{2} < 2 - 1$

Multiply each term in the inequality by $2$ .

$0 \cdot 2 < - \frac{x}{2} \cdot 2 < 2 \cdot 2 - 1 \cdot 2$

Multiply $0$ by $2$ .

$0 < - \frac{x}{2} \cdot 2 < 2 \cdot 2 - 1 \cdot 2$

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{x}{2}$ into the numerator.

$0 < \frac{- x}{2} \cdot 2 < 2 \cdot 2 - 1 \cdot 2$

Cancel the common factor.

$0 < \frac{- x}{2} \cdot 2 < 2 \cdot 2 - 1 \cdot 2$

Rewrite the expression.

$0 < - x < 2 \cdot 2 - 1 \cdot 2$

Simplify each term.

Multiply $2$ by $2$ .

$0 < - x < 4 - 1 \cdot 2$

Multiply $- 1$ by $2$ .

$0 < - x < 4 - 2$

Subtract $2$ from $4$ .

$0 < - x < 2$

Step 3

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

$\frac{0}{- 1} > \frac{- x}{- 1} > \frac{2}{- 1}$

Step 4

Divide $0$ by $- 1$ .

$0 > \frac{- x}{- 1} > \frac{2}{- 1}$

Step 5

Dividing two negative values results in a positive value.

$0 > \frac{x}{1} > \frac{2}{- 1}$

Step 6

Divide $x$ by $1$ .

$0 > x > \frac{2}{- 1}$

Step 7

Divide $2$ by $- 1$ .

$0 > x > - 2$

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.

$- 2 < x < 0$

Step 9

The result can be shown in multiple forms.

Inequality Form:

$- 2 < x < 0$

Interval Notation:

$(- 2, 0)$

Step 10

Solve for x 1<1-1/2x<2

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