Algebra Examples

$35 < 7 (2 - b)$ or $\frac{1}{3} (15 b - 12) \geq 21$

Step 1

Simplify the first inequality.

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Step 1.1

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

$7 (2 - b) > 35$ or $\frac{1}{3} \cdot (15 b - 12) \geq 21$

Step 1.2

Divide each term in $7 (2 - b) > 35$ by $7$ and simplify.

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Step 1.2.1

Divide each term in $7 (2 - b) > 35$ by $7$ .

$\frac{7 (2 - b)}{7} > \frac{35}{7}$ or $\frac{1}{3} \cdot (15 b - 12) \geq 21$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $7$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{7 (2 - b)}{7} > \frac{35}{7}$ or $\frac{1}{3} \cdot (15 b - 12) \geq 21$

Step 1.2.2.1.2

Divide $2 - b$ by $1$ .

$2 - b > \frac{35}{7}$ or $\frac{1}{3} \cdot (15 b - 12) \geq 21$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Divide $35$ by $7$ .

$2 - b > 5$ or $\frac{1}{3} \cdot (15 b - 12) \geq 21$

Step 1.3

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

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Step 1.3.1

Subtract $2$ from both sides of the inequality.

$- b > 5 - 2$ or $\frac{1}{3} \cdot (15 b - 12) \geq 21$

Step 1.3.2

Subtract $2$ from $5$ .

$- b > 3$ or $\frac{1}{3} \cdot (15 b - 12) \geq 21$

Step 1.4

Divide each term in $- b > 3$ by $- 1$ and simplify.

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Step 1.4.1

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

$\frac{- b}{- 1} < \frac{3}{- 1}$ or $\frac{1}{3} \cdot (15 b - 12) \geq 21$

Step 1.4.2

Simplify the left side.

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Step 1.4.2.1

Dividing two negative values results in a positive value.

$\frac{b}{1} < \frac{3}{- 1}$ or $\frac{1}{3} \cdot (15 b - 12) \geq 21$

Step 1.4.2.2

Divide $b$ by $1$ .

$b < \frac{3}{- 1}$ or $\frac{1}{3} \cdot (15 b - 12) \geq 21$

Step 1.4.3

Simplify the right side.

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Step 1.4.3.1

Divide $3$ by $- 1$ .

$b < - 3$ or $\frac{1}{3} \cdot (15 b - 12) \geq 21$

Step 2

Simplify the second inequality.

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Step 2.1

Simplify $\frac{1}{3} \cdot (15 b - 12)$ .

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Step 2.1.1

Apply the distributive property.

$b < - 3$ or $\frac{1}{3} \cdot (15 b) + \frac{1}{3} \cdot - 12 \geq 21$

Step 2.1.2

Cancel the common factor of $3$ .

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Step 2.1.2.1

Factor $3$ out of $15 b$ .

$b < - 3$ or $\frac{1}{3} \cdot (3 (5 b)) + \frac{1}{3} \cdot - 12 \geq 21$

Step 2.1.2.2

Cancel the common factor.

$b < - 3$ or $\frac{1}{3} \cdot (3 (5 b)) + \frac{1}{3} \cdot - 12 \geq 21$

Step 2.1.2.3

Rewrite the expression.

$b < - 3$ or $5 b + \frac{1}{3} \cdot - 12 \geq 21$

Step 2.1.3

Cancel the common factor of $3$ .

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Step 2.1.3.1

Factor $3$ out of $- 12$ .

$b < - 3$ or $5 b + \frac{1}{3} \cdot (3 (- 4)) \geq 21$

Step 2.1.3.2

Cancel the common factor.

$b < - 3$ or $5 b + \frac{1}{3} \cdot (3 \cdot - 4) \geq 21$

Step 2.1.3.3

Rewrite the expression.

$b < - 3$ or $5 b - 4 \geq 21$

Step 2.2

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

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Step 2.2.1

Add $4$ to both sides of the inequality.

$b < - 3$ or $5 b \geq 21 + 4$

Step 2.2.2

Add $21$ and $4$ .

$b < - 3$ or $5 b \geq 25$

Step 2.3

Divide each term in $5 b \geq 25$ by $5$ and simplify.

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Step 2.3.1

Divide each term in $5 b \geq 25$ by $5$ .

$b < - 3$ or $\frac{5 b}{5} \geq \frac{25}{5}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $5$ .

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Step 2.3.2.1.1

Cancel the common factor.

$b < - 3$ or $\frac{5 b}{5} \geq \frac{25}{5}$

Step 2.3.2.1.2

Divide $b$ by $1$ .

$b < - 3$ or $b \geq \frac{25}{5}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Divide $25$ by $5$ .

$b < - 3$ or $b \geq 5$

Step 3

The result can be shown in multiple forms.

Inequality Form:

$b < - 3 or b \geq 5$

Interval Notation:

$(- \infty, - 3) \cup [5, \infty)$

Step 4