Algebra Examples

$2 n + 19 \leq 10 + n$ or $- 3 n + 3 < - 2 n + 33$

Step 1

Simplify the first inequality.

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Step 1.1

Move all terms containing $n$ to the left side of the inequality.

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Step 1.1.1

Subtract $n$ from both sides of the inequality.

$2 n + 19 - n \leq 10$ or $- 3 n + 3 < - 2 n + 33$

Step 1.1.2

Subtract $n$ from $2 n$ .

$n + 19 \leq 10$ or $- 3 n + 3 < - 2 n + 33$

Step 1.2

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

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Step 1.2.1

Subtract $19$ from both sides of the inequality.

$n \leq 10 - 19$ or $- 3 n + 3 < - 2 n + 33$

Step 1.2.2

Subtract $19$ from $10$ .

$n \leq - 9$ or $- 3 n + 3 < - 2 n + 33$

Step 2

Simplify the second inequality.

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Step 2.1

Move all terms containing $n$ to the left side of the inequality.

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Step 2.1.1

Add $2 n$ to both sides of the inequality.

$n \leq - 9$ or $- 3 n + 3 + 2 n < 33$

Step 2.1.2

Add $- 3 n$ and $2 n$ .

$n \leq - 9$ or $- n + 3 < 33$

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 $3$ from both sides of the inequality.

$n \leq - 9$ or $- n < 33 - 3$

Step 2.2.2

Subtract $3$ from $33$ .

$n \leq - 9$ or $- n < 30$

Step 2.3

Divide each term in $- n < 30$ by $- 1$ and simplify.

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Step 2.3.1

Divide each term in $- n < 30$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$n \leq - 9$ or $\frac{- n}{- 1} > \frac{30}{- 1}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Dividing two negative values results in a positive value.

$n \leq - 9$ or $\frac{n}{1} > \frac{30}{- 1}$

Step 2.3.2.2

Divide $n$ by $1$ .

$n \leq - 9$ or $n > \frac{30}{- 1}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Divide $30$ by $- 1$ .

$n \leq - 9$ or $n > - 30$

Step 3

The union consists of all of the elements that are contained in each interval.

All real numbers

Step 4

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$

Step 5