Algebra Examples

$10 \leq - \frac{2}{3} (9 + 12 b)$

Step 1

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

$- \frac{2}{3} \cdot (9 + 12 b) \geq 10$

Step 2

Multiply each term in $- \frac{2}{3} \cdot (9 + 12 b) \geq 10$ by $- \frac{3}{2}$ to eliminate the fractions.

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Step 2.1

Multiply each term in $- \frac{2}{3} \cdot (9 + 12 b) \geq 10$ by $- \frac{3}{2}$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$- \frac{2}{3} \cdot (9 + 12 b) (- \frac{3}{2}) \leq 10 (- \frac{3}{2})$

Step 2.2

Simplify the left side.

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Step 2.2.1

Apply the distributive property.

$(- \frac{2}{3} \cdot 9 - \frac{2}{3} (12 b)) (- \frac{3}{2}) \leq 10 (- \frac{3}{2})$

Step 2.2.2

Cancel the common factor of $3$ .

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Step 2.2.2.1

Move the leading negative in $- \frac{2}{3}$ into the numerator.

$(\frac{- 2}{3} \cdot 9 - \frac{2}{3} (12 b)) (- \frac{3}{2}) \leq 10 (- \frac{3}{2})$

Step 2.2.2.2

Factor $3$ out of $9$ .

$(\frac{- 2}{3} \cdot (3 (3)) - \frac{2}{3} (12 b)) (- \frac{3}{2}) \leq 10 (- \frac{3}{2})$

Step 2.2.2.3

Cancel the common factor.

$(\frac{- 2}{3} \cdot (3 \cdot 3) - \frac{2}{3} (12 b)) (- \frac{3}{2}) \leq 10 (- \frac{3}{2})$

Step 2.2.2.4

Rewrite the expression.

$(- 2 \cdot 3 - \frac{2}{3} (12 b)) (- \frac{3}{2}) \leq 10 (- \frac{3}{2})$

Step 2.2.3

Multiply $- 2$ by $3$ .

$(- 6 - \frac{2}{3} (12 b)) (- \frac{3}{2}) \leq 10 (- \frac{3}{2})$

Step 2.2.4

Cancel the common factor of $3$ .

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Step 2.2.4.1

Move the leading negative in $- \frac{2}{3}$ into the numerator.

$(- 6 + \frac{- 2}{3} (12 b)) (- \frac{3}{2}) \leq 10 (- \frac{3}{2})$

Step 2.2.4.2

Factor $3$ out of $12 b$ .

$(- 6 + \frac{- 2}{3} (3 (4 b))) (- \frac{3}{2}) \leq 10 (- \frac{3}{2})$

Step 2.2.4.3

Cancel the common factor.

$(- 6 + \frac{- 2}{3} (3 (4 b))) (- \frac{3}{2}) \leq 10 (- \frac{3}{2})$

Step 2.2.4.4

Rewrite the expression.

$(- 6 - 2 (4 b)) (- \frac{3}{2}) \leq 10 (- \frac{3}{2})$

Step 2.2.5

Multiply $4$ by $- 2$ .

$(- 6 - 8 b) (- \frac{3}{2}) \leq 10 (- \frac{3}{2})$

Step 2.2.6

Apply the distributive property.

$- 6 (- \frac{3}{2}) - 8 b (- \frac{3}{2}) \leq 10 (- \frac{3}{2})$

Step 2.2.7

Cancel the common factor of $2$ .

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Step 2.2.7.1

Move the leading negative in $- \frac{3}{2}$ into the numerator.

$- 6 (\frac{- 3}{2}) - 8 b (- \frac{3}{2}) \leq 10 (- \frac{3}{2})$

Step 2.2.7.2

Factor $2$ out of $- 6$ .

$2 (- 3) \frac{- 3}{2} - 8 b (- \frac{3}{2}) \leq 10 (- \frac{3}{2})$

Step 2.2.7.3

Cancel the common factor.

$2 \cdot - 3 \frac{- 3}{2} - 8 b (- \frac{3}{2}) \leq 10 (- \frac{3}{2})$

Step 2.2.7.4

Rewrite the expression.

$- 3 \cdot - 3 - 8 b (- \frac{3}{2}) \leq 10 (- \frac{3}{2})$

Step 2.2.8

Multiply $- 3$ by $- 3$ .

$9 - 8 b (- \frac{3}{2}) \leq 10 (- \frac{3}{2})$

Step 2.2.9

Cancel the common factor of $2$ .

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Step 2.2.9.1

Move the leading negative in $- \frac{3}{2}$ into the numerator.

$9 - 8 b \frac{- 3}{2} \leq 10 (- \frac{3}{2})$

Step 2.2.9.2

Factor $2$ out of $- 8 b$ .

$9 + 2 (- 4 b) \frac{- 3}{2} \leq 10 (- \frac{3}{2})$

Step 2.2.9.3

Cancel the common factor.

$9 + 2 (- 4 b) \frac{- 3}{2} \leq 10 (- \frac{3}{2})$

Step 2.2.9.4

Rewrite the expression.

$9 - 4 b \cdot - 3 \leq 10 (- \frac{3}{2})$

Step 2.2.10

Multiply $- 3$ by $- 4$ .

$9 + 12 b \leq 10 (- \frac{3}{2})$

Step 2.3

Simplify the right side.

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Step 2.3.1

Cancel the common factor of $2$ .

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Step 2.3.1.1

Move the leading negative in $- \frac{3}{2}$ into the numerator.

$9 + 12 b \leq 10 (\frac{- 3}{2})$

Step 2.3.1.2

Factor $2$ out of $10$ .

$9 + 12 b \leq 2 (5) \frac{- 3}{2}$

Step 2.3.1.3

Cancel the common factor.

$9 + 12 b \leq 2 \cdot 5 \frac{- 3}{2}$

Step 2.3.1.4

Rewrite the expression.

$9 + 12 b \leq 5 \cdot - 3$

Step 2.3.2

Multiply $5$ by $- 3$ .

$9 + 12 b \leq - 15$

Step 3

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

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Step 3.1

Subtract $9$ from both sides of the inequality.

$12 b \leq - 15 - 9$

Step 3.2

Subtract $9$ from $- 15$ .

$12 b \leq - 24$

Step 4

Divide each term in $12 b \leq - 24$ by $12$ and simplify.

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Step 4.1

Divide each term in $12 b \leq - 24$ by $12$ .

$\frac{12 b}{12} \leq \frac{- 24}{12}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $12$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{12 b}{12} \leq \frac{- 24}{12}$

Step 4.2.1.2

Divide $b$ by $1$ .

$b \leq \frac{- 24}{12}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $- 24$ by $12$ .

$b \leq - 2$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$b \leq - 2$

Interval Notation:

$(- \infty, - 2]$