Algebra Examples

$\frac{x - 2}{x + 3} \leq \frac{5}{3}$

Step 1

Subtract $\frac{5}{3}$ from both sides of the inequality.

$\frac{x - 2}{x + 3} - \frac{5}{3} \leq 0$

Step 2

Simplify $\frac{x - 2}{x + 3} - \frac{5}{3}$ .

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Step 2.1

To write $\frac{x - 2}{x + 3}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{x - 2}{x + 3} \cdot \frac{3}{3} - \frac{5}{3} \leq 0$

Step 2.2

To write $- \frac{5}{3}$ as a fraction with a common denominator, multiply by $\frac{x + 3}{x + 3}$ .

$\frac{x - 2}{x + 3} \cdot \frac{3}{3} - \frac{5}{3} \cdot \frac{x + 3}{x + 3} \leq 0$

Step 2.3

Write each expression with a common denominator of $(x + 3) \cdot 3$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.3.1

Multiply $\frac{x - 2}{x + 3}$ by $\frac{3}{3}$ .

$\frac{(x - 2) \cdot 3}{(x + 3) \cdot 3} - \frac{5}{3} \cdot \frac{x + 3}{x + 3} \leq 0$

Step 2.3.2

Multiply $\frac{5}{3}$ by $\frac{x + 3}{x + 3}$ .

$\frac{(x - 2) \cdot 3}{(x + 3) \cdot 3} - \frac{5 (x + 3)}{3 (x + 3)} \leq 0$

Step 2.3.3

Reorder the factors of $(x + 3) \cdot 3$ .

$\frac{(x - 2) \cdot 3}{3 (x + 3)} - \frac{5 (x + 3)}{3 (x + 3)} \leq 0$

Step 2.4

Combine the numerators over the common denominator.

$\frac{(x - 2) \cdot 3 - 5 (x + 3)}{3 (x + 3)} \leq 0$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Apply the distributive property.

$\frac{x \cdot 3 - 2 \cdot 3 - 5 (x + 3)}{3 (x + 3)} \leq 0$

Step 2.5.2

Move $3$ to the left of $x$ .

$\frac{3 \cdot x - 2 \cdot 3 - 5 (x + 3)}{3 (x + 3)} \leq 0$

Step 2.5.3

Multiply $- 2$ by $3$ .

$\frac{3 \cdot x - 6 - 5 (x + 3)}{3 (x + 3)} \leq 0$

Step 2.5.4

Apply the distributive property.

$\frac{3 x - 6 - 5 x - 5 \cdot 3}{3 (x + 3)} \leq 0$

Step 2.5.5

Multiply $- 5$ by $3$ .

$\frac{3 x - 6 - 5 x - 15}{3 (x + 3)} \leq 0$

Step 2.5.6

Subtract $5 x$ from $3 x$ .

$\frac{- 2 x - 6 - 15}{3 (x + 3)} \leq 0$

Step 2.5.7

Subtract $15$ from $- 6$ .

$\frac{- 2 x - 21}{3 (x + 3)} \leq 0$

Step 2.6

Factor $- 1$ out of $- 2 x$ .

$\frac{- (2 x) - 21}{3 (x + 3)} \leq 0$

Step 2.7

Rewrite $- 21$ as $- 1 (21)$ .

$\frac{- (2 x) - 1 (21)}{3 (x + 3)} \leq 0$

Step 2.8

Factor $- 1$ out of $- (2 x) - 1 (21)$ .

$\frac{- (2 x + 21)}{3 (x + 3)} \leq 0$

Step 2.9

Rewrite $- (2 x + 21)$ as $- 1 (2 x + 21)$ .

$\frac{- 1 (2 x + 21)}{3 (x + 3)} \leq 0$

Step 2.10

Move the negative in front of the fraction.

$- \frac{2 x + 21}{3 (x + 3)} \leq 0$

Step 3

Find all the values where the expression switches from negative to positive by setting each factor equal to $0$ and solving.

$2 x + 21 = 0$

$x + 3 = 0$

Step 4

Subtract $21$ from both sides of the equation.

$2 x = - 21$

Step 5

Divide each term in $2 x = - 21$ by $2$ and simplify.

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Step 5.1

Divide each term in $2 x = - 21$ by $2$ .

$\frac{2 x}{2} = \frac{- 21}{2}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $2$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 21}{2}$

Step 5.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 21}{2}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Move the negative in front of the fraction.

$x = - \frac{21}{2}$

Step 6

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 7

Solve for each factor to find the values where the absolute value expression goes from negative to positive.

$x = - \frac{21}{2}$

$x = - 3$

Step 8

Consolidate the solutions.

$x = - \frac{21}{2}, - 3$

Step 9

Find the domain of $- \frac{2 x + 21}{3 (x + 3)}$ .

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Step 9.1

Set the denominator in $\frac{2 x + 21}{3 (x + 3)}$ equal to $0$ to find where the expression is undefined.

$3 (x + 3) = 0$

Step 9.2

Solve for $x$ .

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Step 9.2.1

Divide each term in $3 (x + 3) = 0$ by $3$ and simplify.

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Step 9.2.1.1

Divide each term in $3 (x + 3) = 0$ by $3$ .

$\frac{3 (x + 3)}{3} = \frac{0}{3}$

Step 9.2.1.2

Simplify the left side.

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Step 9.2.1.2.1

Cancel the common factor of $3$ .

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Step 9.2.1.2.1.1

Cancel the common factor.

$\frac{3 (x + 3)}{3} = \frac{0}{3}$

Step 9.2.1.2.1.2

Divide $x + 3$ by $1$ .

$x + 3 = \frac{0}{3}$

Step 9.2.1.3

Simplify the right side.

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Step 9.2.1.3.1

Divide $0$ by $3$ .

$x + 3 = 0$

Step 9.2.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 9.3

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

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

Step 10

Use each root to create test intervals.

$x < - \frac{21}{2}$

$- \frac{21}{2} < x < - 3$

$x > - 3$

Step 11

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 11.1

Test a value on the interval $x < - \frac{21}{2}$ to see if it makes the inequality true.

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Step 11.1.1

Choose a value on the interval $x < - \frac{21}{2}$ and see if this value makes the original inequality true.

$x = - 13$

Step 11.1.2

Replace $x$ with $- 13$ in the original inequality.

$\frac{(- 13) - 2}{(- 13) + 3} \leq \frac{5}{3}$

Step 11.1.3

The left side $1.5$ is less than the right side $1.\overline{6}$ , which means that the given statement is always true.

True

Step 11.2

Test a value on the interval $- \frac{21}{2} < x < - 3$ to see if it makes the inequality true.

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Step 11.2.1

Choose a value on the interval $- \frac{21}{2} < x < - 3$ and see if this value makes the original inequality true.

$x = - 7$

Step 11.2.2

Replace $x$ with $- 7$ in the original inequality.

$\frac{(- 7) - 2}{(- 7) + 3} \leq \frac{5}{3}$

Step 11.2.3

The left side $2.25$ is greater than the right side $1.\overline{6}$ , which means that the given statement is false.

False

Step 11.3

Test a value on the interval $x > - 3$ to see if it makes the inequality true.

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Step 11.3.1

Choose a value on the interval $x > - 3$ and see if this value makes the original inequality true.

$x = 0$

Step 11.3.2

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

$\frac{(0) - 2}{(0) + 3} \leq \frac{5}{3}$

Step 11.3.3

The left side $- 0.\overline{6}$ is less than the right side $1.\overline{6}$ , which means that the given statement is always true.

True

Step 11.4

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

$x < - \frac{21}{2}$ True

$- \frac{21}{2} < x < - 3$ False

$x > - 3$ True

$x < - \frac{21}{2}$ True

$- \frac{21}{2} < x < - 3$ False

$x > - 3$ True

Step 12

The solution consists of all of the true intervals.

$x \leq - \frac{21}{2}$ or $x > - 3$

Step 13

The result can be shown in multiple forms.

Inequality Form:

$x \leq - \frac{21}{2} or x > - 3$

Interval Notation:

$(- \infty, - \frac{21}{2}] \cup (- 3, \infty)$

Step 14