Precalculus Examples

$9 \leq \frac{3}{2} (6 - 3 x) < 27$

Step 1

Apply the distributive property.

$9 \leq \frac{3}{2} \cdot 6 + \frac{3}{2} \cdot (- 3 x) < 27$

Step 2

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.1

Factor $2$ out of $6$ .

$9 \leq \frac{3}{2} \cdot (2 (3)) + \frac{3}{2} \cdot (- 3 x) < 27$

Step 2.2

Cancel the common factor.

$9 \leq \frac{3}{2} \cdot (2 \cdot 3) + \frac{3}{2} \cdot (- 3 x) < 27$

Step 2.3

Rewrite the expression.

$9 \leq 3 \cdot 3 + \frac{3}{2} \cdot (- 3 x) < 27$

Step 3

Multiply $3$ by $3$ .

$9 \leq 9 + \frac{3}{2} \cdot (- 3 x) < 27$

Step 4

Multiply $\frac{3}{2} (- 3 x)$ .

Tap for more steps...

Step 4.1

Combine $- 3$ and $\frac{3}{2}$ .

$9 \leq 9 + \frac{- 3 \cdot 3}{2} x < 27$

Step 4.2

Multiply $- 3$ by $3$ .

$9 \leq 9 + \frac{- 9}{2} x < 27$

Step 4.3

Combine $\frac{- 9}{2}$ and $x$ .

$9 \leq 9 + \frac{- 9 x}{2} < 27$

Step 5

Move the negative in front of the fraction.

$9 \leq 9 - \frac{9 x}{2} < 27$

Step 6

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

Tap for more steps...

Step 6.1

Subtract $9$ from each section of the inequality because it does not contain the variable we are trying to solve for.

$9 - 9 \leq - \frac{9 x}{2} < 27 - 9$

Step 6.2

Subtract $9$ from $9$ .

$0 \leq - \frac{9 x}{2} < 27 - 9$

Step 6.3

Multiply each term in the inequality by $2$ .

$0 \cdot 2 \leq - \frac{9 x}{2} \cdot 2 < 27 \cdot 2 - 9 \cdot 2$

Step 6.4

Multiply $0$ by $2$ .

$0 \leq - \frac{9 x}{2} \cdot 2 < 27 \cdot 2 - 9 \cdot 2$

Step 6.5

Cancel the common factor of $2$ .

Tap for more steps...

Step 6.5.1

Move the leading negative in $- \frac{9 x}{2}$ into the numerator.

$0 \leq \frac{- 9 x}{2} \cdot 2 < 27 \cdot 2 - 9 \cdot 2$

Step 6.5.2

Cancel the common factor.

$0 \leq \frac{- 9 x}{2} \cdot 2 < 27 \cdot 2 - 9 \cdot 2$

Step 6.5.3

Rewrite the expression.

$0 \leq - 9 x < 27 \cdot 2 - 9 \cdot 2$

Step 6.6

Simplify each term.

Tap for more steps...

Step 6.6.1

Multiply $27$ by $2$ .

$0 \leq - 9 x < 54 - 9 \cdot 2$

Step 6.6.2

Multiply $- 9$ by $2$ .

$0 \leq - 9 x < 54 - 18$

Step 6.7

Subtract $18$ from $54$ .

$0 \leq - 9 x < 36$

Step 7

Divide each term in the inequality by $- 9$ .

$\frac{0}{- 9} \geq \frac{- 9 x}{- 9} > \frac{36}{- 9}$

Step 8

Divide $0$ by $- 9$ .

$0 \geq \frac{- 9 x}{- 9} > \frac{36}{- 9}$

Step 9

Cancel the common factor of $- 9$ .

Tap for more steps...

Step 9.1

Cancel the common factor.

$0 \geq \frac{- 9 x}{- 9} > \frac{36}{- 9}$

Step 9.2

Divide $x$ by $1$ .

$0 \geq x > \frac{36}{- 9}$

Step 10

Divide $36$ by $- 9$ .

$0 \geq x > - 4$

Step 11

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

$- 4 < x \leq 0$

Step 12

The result can be shown in multiple forms.

Inequality Form:

$- 4 < x \leq 0$

Interval Notation:

$(- 4, 0]$

Step 13