Algebra Examples

$7 - n \geq - 6$ and $2 (3 n + 8) \geq 10$

Step 1

Simplify the first inequality.

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Step 1.1

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

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Step 1.1.1

Subtract $7$ from both sides of the inequality.

$- n \geq - 6 - 7$

Step 1.1.2

Subtract $7$ from $- 6$ .

$- n \geq - 13$

Step 1.2

Divide each term in $- n \geq - 13$ by $- 1$ and simplify.

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Step 1.2.1

Divide each term in $- n \geq - 13$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- n}{- 1} \leq \frac{- 13}{- 1}$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Dividing two negative values results in a positive value.

$\frac{n}{1} \leq \frac{- 13}{- 1}$

Step 1.2.2.2

Divide $n$ by $1$ .

$n \leq \frac{- 13}{- 1}$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Divide $- 13$ by $- 1$ .

$n \leq 13$

Step 2

Simplify the second inequality.

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Step 2.1

Divide each term in $2 (3 n + 8) \geq 10$ by $2$ and simplify.

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Step 2.1.1

Divide each term in $2 (3 n + 8) \geq 10$ by $2$ .

$\frac{2 (3 n + 8)}{2} \geq \frac{10}{2}$

Step 2.1.2

Simplify the left side.

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Step 2.1.2.1

Cancel the common factor of $2$ .

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Step 2.1.2.1.1

Cancel the common factor.

$\frac{2 (3 n + 8)}{2} \geq \frac{10}{2}$

Step 2.1.2.1.2

Divide $3 n + 8$ by $1$ .

$3 n + 8 \geq \frac{10}{2}$

Step 2.1.3

Simplify the right side.

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Step 2.1.3.1

Divide $10$ by $2$ .

$3 n + 8 \geq 5$

Step 2.2

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

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Step 2.2.1

Subtract $8$ from both sides of the inequality.

$3 n \geq 5 - 8$

Step 2.2.2

Subtract $8$ from $5$ .

$3 n \geq - 3$

Step 2.3

Divide each term in $3 n \geq - 3$ by $3$ and simplify.

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Step 2.3.1

Divide each term in $3 n \geq - 3$ by $3$ .

$\frac{3 n}{3} \geq \frac{- 3}{3}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $3$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{3 n}{3} \geq \frac{- 3}{3}$

Step 2.3.2.1.2

Divide $n$ by $1$ .

$n \geq \frac{- 3}{3}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Divide $- 3$ by $3$ .

$n \geq - 1$

Step 3

The intersection consists of the elements that are contained in both intervals.

$- 1 \leq n \leq 13$

Step 4

Convert the inequality to interval notation.

$[- 1, 13]$

Step 5