Algebra Examples

$\frac{2 n + 1}{3 n + 1} \leq \frac{n - 1}{3 n + 1}$

Step 1

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$2 n + 1 \leq n - 1$

Step 2

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

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Step 2.1

Subtract $n$ from both sides of the inequality.

$2 n + 1 - n \leq - 1$

Step 2.2

Subtract $n$ from $2 n$ .

$n + 1 \leq - 1$

Step 3

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

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Step 3.1

Subtract $1$ from both sides of the inequality.

$n \leq - 1 - 1$

Step 3.2

Subtract $1$ from $- 1$ .

$n \leq - 2$

Step 4

Find the domain of $\frac{n + 2}{3 n + 1}$ .

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Step 4.1

Set the denominator in $\frac{n + 2}{3 n + 1}$ equal to $0$ to find where the expression is undefined.

$3 n + 1 = 0$

Step 4.2

Solve for $n$ .

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Step 4.2.1

Subtract $1$ from both sides of the equation.

$3 n = - 1$

Step 4.2.2

Divide each term in $3 n = - 1$ by $3$ and simplify.

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Step 4.2.2.1

Divide each term in $3 n = - 1$ by $3$ .

$\frac{3 n}{3} = \frac{- 1}{3}$

Step 4.2.2.2

Simplify the left side.

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Step 4.2.2.2.1

Cancel the common factor of $3$ .

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Step 4.2.2.2.1.1

Cancel the common factor.

$\frac{3 n}{3} = \frac{- 1}{3}$

Step 4.2.2.2.1.2

Divide $n$ by $1$ .

$n = \frac{- 1}{3}$

Step 4.2.2.3

Simplify the right side.

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Step 4.2.2.3.1

Move the negative in front of the fraction.

$n = - \frac{1}{3}$

Step 4.3

The domain is all values of $n$ that make the expression defined.

$(- \infty, - \frac{1}{3}) \cup (- \frac{1}{3}, \infty)$

Step 5

Use each root to create test intervals.

$n < - 2$

$- 2 < n < - \frac{1}{3}$

$n > - \frac{1}{3}$

Step 6

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

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Step 6.1

Test a value on the interval $n < - 2$ to see if it makes the inequality true.

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Step 6.1.1

Choose a value on the interval $n < - 2$ and see if this value makes the original inequality true.

$n = - 4$

Step 6.1.2

Replace $n$ with $- 4$ in the original inequality.

$\frac{2 (- 4) + 1}{3 (- 4) + 1} \leq \frac{(- 4) - 1}{3 (- 4) + 1}$

Step 6.1.3

The left side $0.\overline{63}$ is greater than the right side $0.\overline{45}$ , which means that the given statement is false.

False

Step 6.2

Test a value on the interval $- 2 < n < - \frac{1}{3}$ to see if it makes the inequality true.

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Step 6.2.1

Choose a value on the interval $- 2 < n < - \frac{1}{3}$ and see if this value makes the original inequality true.

$n = - 1$

Step 6.2.2

Replace $n$ with $- 1$ in the original inequality.

$\frac{2 (- 1) + 1}{3 (- 1) + 1} \leq \frac{(- 1) - 1}{3 (- 1) + 1}$

Step 6.2.3

The left side $0.5$ is less than the right side $1$ , which means that the given statement is always true.

True

Step 6.3

Test a value on the interval $n > - \frac{1}{3}$ to see if it makes the inequality true.

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Step 6.3.1

Choose a value on the interval $n > - \frac{1}{3}$ and see if this value makes the original inequality true.

$n = 0$

Step 6.3.2

Replace $n$ with $0$ in the original inequality.

$\frac{2 (0) + 1}{3 (0) + 1} \leq \frac{(0) - 1}{3 (0) + 1}$

Step 6.3.3

The left side $1$ is greater than the right side $- 1$ , which means that the given statement is false.

False

Step 6.4

Compare the intervals to determine which ones satisfy the original inequality.

$n < - 2$ False

$- 2 < n < - \frac{1}{3}$ True

$n > - \frac{1}{3}$ False

$n < - 2$ False

$- 2 < n < - \frac{1}{3}$ True

$n > - \frac{1}{3}$ False

Step 7

The solution consists of all of the true intervals.

$- 2 \leq n < - \frac{1}{3}$

Step 8

The result can be shown in multiple forms.

Inequality Form:

$- 2 \leq n < - \frac{1}{3}$

Interval Notation:

$[- 2, - \frac{1}{3})$

Step 9