Precalculus Examples

$- 1 < 2 - \frac{x}{3} < 1$

Step 1

Solve $- 1 < 2 - \frac{x}{3}$ for $x$ .

Tap for more steps...

Step 1.1

Rewrite so $x$ is on the left side of the inequality.

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

Step 1.2

Move all terms not containing $x$ to the right side of the inequality.

Tap for more steps...

Step 1.2.1

Subtract $2$ from both sides of the inequality.

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

Step 1.2.2

Subtract $2$ from $- 1$ .

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

Step 1.3

Divide each term in $- \frac{x}{3} > - 3$ by $- 1$ and simplify.

Tap for more steps...

Step 1.3.1

Divide each term in $- \frac{x}{3} > - 3$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

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

Step 1.3.2

Simplify the left side.

Tap for more steps...

Step 1.3.2.1

Dividing two negative values results in a positive value.

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

Step 1.3.2.2

Divide $\frac{x}{3}$ by $1$ .

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

Step 1.3.3

Simplify the right side.

Tap for more steps...

Step 1.3.3.1

Divide $- 3$ by $- 1$ .

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

Step 1.4

Multiply both sides by $3$ .

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

Step 1.5

Simplify.

Tap for more steps...

Step 1.5.1

Simplify the left side.

Tap for more steps...

Step 1.5.1.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 1.5.1.1.1

Cancel the common factor.

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

Step 1.5.1.1.2

Rewrite the expression.

$x < 3 \cdot 3$

Step 1.5.2

Simplify the right side.

Tap for more steps...

Step 1.5.2.1

Multiply $3$ by $3$ .

$x < 9$

Step 2

Solve $2 - \frac{x}{3} < 1$ for $x$ .

Tap for more steps...

Step 2.1

Move all terms not containing $x$ to the right side of the inequality.

Tap for more steps...

Step 2.1.1

Subtract $2$ from both sides of the inequality.

$- \frac{x}{3} < 1 - 2$

Step 2.1.2

Subtract $2$ from $1$ .

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

Step 2.2

Divide each term in $- \frac{x}{3} < - 1$ by $- 1$ and simplify.

Tap for more steps...

Step 2.2.1

Divide each term in $- \frac{x}{3} < - 1$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

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

Step 2.2.2

Simplify the left side.

Tap for more steps...

Step 2.2.2.1

Dividing two negative values results in a positive value.

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

Step 2.2.2.2

Divide $\frac{x}{3}$ by $1$ .

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

Step 2.2.3

Simplify the right side.

Tap for more steps...

Step 2.2.3.1

Divide $- 1$ by $- 1$ .

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

Step 2.3

Multiply both sides by $3$ .

$\frac{x}{3} \cdot 3 > 1 \cdot 3$

Step 2.4

Simplify.

Tap for more steps...

Step 2.4.1

Simplify the left side.

Tap for more steps...

Step 2.4.1.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 2.4.1.1.1

Cancel the common factor.

$\frac{x}{3} \cdot 3 > 1 \cdot 3$

Step 2.4.1.1.2

Rewrite the expression.

$x > 1 \cdot 3$

Step 2.4.2

Simplify the right side.

Tap for more steps...

Step 2.4.2.1

Multiply $3$ by $1$ .

$x > 3$

Step 3

Find the intersection of $x < 9$ and $x > 3$ .

$3 < x < 9$

Step 4

The result can be shown in multiple forms.

Inequality Form:

$3 < x < 9$

Interval Notation:

$(3, 9)$

Step 5