Finite Math Examples

$\frac{5}{t} < \frac{9}{2 t + 1}$

Step 1

Subtract $\frac{9}{2 t + 1}$ from both sides of the inequality.

$\frac{5}{t} - \frac{9}{2 t + 1} < 0$

Step 2

Simplify $\frac{5}{t} - \frac{9}{2 t + 1}$ .

Tap for more steps...

Step 2.1

To write $\frac{5}{t}$ as a fraction with a common denominator, multiply by $\frac{2 t + 1}{2 t + 1}$ .

$\frac{5}{t} \cdot \frac{2 t + 1}{2 t + 1} - \frac{9}{2 t + 1} < 0$

Step 2.2

To write $- \frac{9}{2 t + 1}$ as a fraction with a common denominator, multiply by $\frac{t}{t}$ .

$\frac{5}{t} \cdot \frac{2 t + 1}{2 t + 1} - \frac{9}{2 t + 1} \cdot \frac{t}{t} < 0$

Step 2.3

Write each expression with a common denominator of $t (2 t + 1)$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 2.3.1

Multiply $\frac{5}{t}$ by $\frac{2 t + 1}{2 t + 1}$ .

$\frac{5 (2 t + 1)}{t (2 t + 1)} - \frac{9}{2 t + 1} \cdot \frac{t}{t} < 0$

Step 2.3.2

Multiply $\frac{9}{2 t + 1}$ by $\frac{t}{t}$ .

$\frac{5 (2 t + 1)}{t (2 t + 1)} - \frac{9 t}{(2 t + 1) t} < 0$

Step 2.3.3

Reorder the factors of $(2 t + 1) t$ .

$\frac{5 (2 t + 1)}{t (2 t + 1)} - \frac{9 t}{t (2 t + 1)} < 0$

Step 2.4

Combine the numerators over the common denominator.

$\frac{5 (2 t + 1) - 9 t}{t (2 t + 1)} < 0$

Step 2.5

Simplify the numerator.

Tap for more steps...

Step 2.5.1

Apply the distributive property.

$\frac{5 (2 t) + 5 \cdot 1 - 9 t}{t (2 t + 1)} < 0$

Step 2.5.2

Multiply $2$ by $5$ .

$\frac{10 t + 5 \cdot 1 - 9 t}{t (2 t + 1)} < 0$

Step 2.5.3

Multiply $5$ by $1$ .

$\frac{10 t + 5 - 9 t}{t (2 t + 1)} < 0$

Step 2.5.4

Subtract $9 t$ from $10 t$ .

$\frac{t + 5}{t (2 t + 1)} < 0$

Step 3

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

$t = 0$

$t + 5 = 0$

$2 t + 1 = 0$

Step 4

Subtract $5$ from both sides of the equation.

$t = - 5$

Step 5

Subtract $1$ from both sides of the equation.

$2 t = - 1$

Step 6

Divide each term in $2 t = - 1$ by $2$ and simplify.

Tap for more steps...

Step 6.1

Divide each term in $2 t = - 1$ by $2$ .

$\frac{2 t}{2} = \frac{- 1}{2}$

Step 6.2

Simplify the left side.

Tap for more steps...

Step 6.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 6.2.1.1

Cancel the common factor.

$\frac{2 t}{2} = \frac{- 1}{2}$

Step 6.2.1.2

Divide $t$ by $1$ .

$t = \frac{- 1}{2}$

Step 6.3

Simplify the right side.

Tap for more steps...

Step 6.3.1

Move the negative in front of the fraction.

$t = - \frac{1}{2}$

Step 7

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

$t = 0$

$t = - 5$

$t = - \frac{1}{2}$

Step 8

Consolidate the solutions.

$t = 0, - 5, - \frac{1}{2}$

Step 9

Find the domain of $\frac{t + 5}{t (2 t + 1)}$ .

Tap for more steps...

Step 9.1

Set the denominator in $\frac{t + 5}{t (2 t + 1)}$ equal to $0$ to find where the expression is undefined.

$t (2 t + 1) = 0$

Step 9.2

Solve for $t$ .

Tap for more steps...

Step 9.2.1

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$t = 0$

$2 t + 1 = 0$

Step 9.2.2

Set $t$ equal to $0$ .

$t = 0$

Step 9.2.3

Set $2 t + 1$ equal to $0$ and solve for $t$ .

Tap for more steps...

Step 9.2.3.1

Set $2 t + 1$ equal to $0$ .

$2 t + 1 = 0$

Step 9.2.3.2

Solve $2 t + 1 = 0$ for $t$ .

Tap for more steps...

Step 9.2.3.2.1

Subtract $1$ from both sides of the equation.

$2 t = - 1$

Step 9.2.3.2.2

Divide each term in $2 t = - 1$ by $2$ and simplify.

Tap for more steps...

Step 9.2.3.2.2.1

Divide each term in $2 t = - 1$ by $2$ .

$\frac{2 t}{2} = \frac{- 1}{2}$

Step 9.2.3.2.2.2

Simplify the left side.

Tap for more steps...

Step 9.2.3.2.2.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 9.2.3.2.2.2.1.1

Cancel the common factor.

$\frac{2 t}{2} = \frac{- 1}{2}$

Step 9.2.3.2.2.2.1.2

Divide $t$ by $1$ .

$t = \frac{- 1}{2}$

Step 9.2.3.2.2.3

Simplify the right side.

Tap for more steps...

Step 9.2.3.2.2.3.1

Move the negative in front of the fraction.

$t = - \frac{1}{2}$

Step 9.2.4

The final solution is all the values that make $t (2 t + 1) = 0$ true.

$t = 0, - \frac{1}{2}$

Step 9.3

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

$(- \infty, - \frac{1}{2}) \cup (- \frac{1}{2}, 0) \cup (0, \infty)$

Step 10

Use each root to create test intervals.

$t < - 5$

$- 5 < t < - \frac{1}{2}$

$- \frac{1}{2} < t < 0$

$t > 0$

Step 11

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

Tap for more steps...

Step 11.1

Test a value on the interval $t < - 5$ to see if it makes the inequality true.

Tap for more steps...

Step 11.1.1

Choose a value on the interval $t < - 5$ and see if this value makes the original inequality true.

$t = - 8$

Step 11.1.2

Replace $t$ with $- 8$ in the original inequality.

$\frac{5}{- 8} < \frac{9}{2 (- 8) + 1}$

Step 11.1.3

The left side $- 0.625$ is less than the right side $- 0.6$ , which means that the given statement is always true.

True

Step 11.2

Test a value on the interval $- 5 < t < - \frac{1}{2}$ to see if it makes the inequality true.

Tap for more steps...

Step 11.2.1

Choose a value on the interval $- 5 < t < - \frac{1}{2}$ and see if this value makes the original inequality true.

$t = - 3$

Step 11.2.2

Replace $t$ with $- 3$ in the original inequality.

$\frac{5}{- 3} < \frac{9}{2 (- 3) + 1}$

Step 11.2.3

The left side $- 1.\overline{6}$ is not less than the right side $- 1.8$ , which means that the given statement is false.

False

Step 11.3

Test a value on the interval $- \frac{1}{2} < t < 0$ to see if it makes the inequality true.

Tap for more steps...

Step 11.3.1

Choose a value on the interval $- \frac{1}{2} < t < 0$ and see if this value makes the original inequality true.

$t = - 0.25$

Step 11.3.2

Replace $t$ with $- 0.25$ in the original inequality.

$\frac{5}{- 0.25} < \frac{9}{2 (- 0.25) + 1}$

Step 11.3.3

The left side $- 20$ is less than the right side $18$ , which means that the given statement is always true.

True

Step 11.4

Test a value on the interval $t > 0$ to see if it makes the inequality true.

Tap for more steps...

Step 11.4.1

Choose a value on the interval $t > 0$ and see if this value makes the original inequality true.

$t = 2$

Step 11.4.2

Replace $t$ with $2$ in the original inequality.

$\frac{5}{2} < \frac{9}{2 (2) + 1}$

Step 11.4.3

The left side $2.5$ is not less than the right side $1.8$ , which means that the given statement is false.

False

Step 11.5

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

$t < - 5$ True

$- 5 < t < - \frac{1}{2}$ False

$- \frac{1}{2} < t < 0$ True

$t > 0$ False

$t < - 5$ True

$- 5 < t < - \frac{1}{2}$ False

$- \frac{1}{2} < t < 0$ True

$t > 0$ False

Step 12

The solution consists of all of the true intervals.

$t < - 5$ or $- \frac{1}{2} < t < 0$

Step 13

The result can be shown in multiple forms.

Inequality Form:

$t < - 5 or - \frac{1}{2} < t < 0$

Interval Notation:

$(- \infty, - 5) \cup (- \frac{1}{2}, 0)$

Step 14