Algebra Examples

$\frac{5}{2 c - 9} > 1$

Step 1

Subtract $1$ from both sides of the inequality.

$\frac{5}{2 c - 9} - 1 > 0$

Step 2

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

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Step 2.1

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

$\frac{5}{2 c - 9} - 1 \cdot \frac{2 c - 9}{2 c - 9} > 0$

Step 2.2

Combine $- 1$ and $\frac{2 c - 9}{2 c - 9}$ .

$\frac{5}{2 c - 9} + \frac{- (2 c - 9)}{2 c - 9} > 0$

Step 2.3

Combine the numerators over the common denominator.

$\frac{5 - (2 c - 9)}{2 c - 9} > 0$

Step 2.4

Simplify the numerator.

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Step 2.4.1

Apply the distributive property.

$\frac{5 - (2 c) - - 9}{2 c - 9} > 0$

Step 2.4.2

Multiply $2$ by $- 1$ .

$\frac{5 - 2 c - - 9}{2 c - 9} > 0$

Step 2.4.3

Multiply $- 1$ by $- 9$ .

$\frac{5 - 2 c + 9}{2 c - 9} > 0$

Step 2.4.4

Add $5$ and $9$ .

$\frac{- 2 c + 14}{2 c - 9} > 0$

Step 2.4.5

Factor $2$ out of $- 2 c + 14$ .

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Step 2.4.5.1

Factor $2$ out of $- 2 c$ .

$\frac{2 (- c) + 14}{2 c - 9} > 0$

Step 2.4.5.2

Factor $2$ out of $14$ .

$\frac{2 (- c) + 2 (7)}{2 c - 9} > 0$

Step 2.4.5.3

Factor $2$ out of $2 (- c) + 2 (7)$ .

$\frac{2 (- c + 7)}{2 c - 9} > 0$

Step 2.5

Factor $- 1$ out of $- c$ .

$\frac{2 (- (c) + 7)}{2 c - 9} > 0$

Step 2.6

Rewrite $7$ as $- 1 (- 7)$ .

$\frac{2 (- (c) - 1 (- 7))}{2 c - 9} > 0$

Step 2.7

Factor $- 1$ out of $- (c) - 1 (- 7)$ .

$\frac{2 (- (c - 7))}{2 c - 9} > 0$

Step 2.8

Rewrite $- (c - 7)$ as $- 1 (c - 7)$ .

$\frac{2 (- 1 (c - 7))}{2 c - 9} > 0$

Step 2.9

Move the negative in front of the fraction.

$- \frac{2 (c - 7)}{2 c - 9} > 0$

Step 3

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

$c - 7 = 0$

$2 c - 9 = 0$

Step 4

Add $7$ to both sides of the equation.

$c = 7$

Step 5

Add $9$ to both sides of the equation.

$2 c = 9$

Step 6

Divide each term in $2 c = 9$ by $2$ and simplify.

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Step 6.1

Divide each term in $2 c = 9$ by $2$ .

$\frac{2 c}{2} = \frac{9}{2}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $2$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{2 c}{2} = \frac{9}{2}$

Step 6.2.1.2

Divide $c$ by $1$ .

$c = \frac{9}{2}$

Step 7

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

$c = 7$

$c = \frac{9}{2}$

Step 8

Consolidate the solutions.

$c = 7, \frac{9}{2}$

Step 9

Find the domain of $- \frac{2 (c - 7)}{2 c - 9}$ .

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Step 9.1

Set the denominator in $\frac{2 (c - 7)}{2 c - 9}$ equal to $0$ to find where the expression is undefined.

$2 c - 9 = 0$

Step 9.2

Solve for $c$ .

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Step 9.2.1

Add $9$ to both sides of the equation.

$2 c = 9$

Step 9.2.2

Divide each term in $2 c = 9$ by $2$ and simplify.

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Step 9.2.2.1

Divide each term in $2 c = 9$ by $2$ .

$\frac{2 c}{2} = \frac{9}{2}$

Step 9.2.2.2

Simplify the left side.

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Step 9.2.2.2.1

Cancel the common factor of $2$ .

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Step 9.2.2.2.1.1

Cancel the common factor.

$\frac{2 c}{2} = \frac{9}{2}$

Step 9.2.2.2.1.2

Divide $c$ by $1$ .

$c = \frac{9}{2}$

Step 9.3

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

$(- \infty, \frac{9}{2}) \cup (\frac{9}{2}, \infty)$

Step 10

Use each root to create test intervals.

$c < \frac{9}{2}$

$\frac{9}{2} < c < 7$

$c > 7$

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 $c < \frac{9}{2}$ to see if it makes the inequality true.

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Step 11.1.1

Choose a value on the interval $c < \frac{9}{2}$ and see if this value makes the original inequality true.

$c = 0$

Step 11.1.2

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

$\frac{5}{2 (0) - 9} > 1$

Step 11.1.3

The left side $- 0.\overline{5}$ is not greater than the right side $1$ , which means that the given statement is false.

False

Step 11.2

Test a value on the interval $\frac{9}{2} < c < 7$ to see if it makes the inequality true.

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Step 11.2.1

Choose a value on the interval $\frac{9}{2} < c < 7$ and see if this value makes the original inequality true.

$c = 6$

Step 11.2.2

Replace $c$ with $6$ in the original inequality.

$\frac{5}{2 (6) - 9} > 1$

Step 11.2.3

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

True

Step 11.3

Test a value on the interval $c > 7$ to see if it makes the inequality true.

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Step 11.3.1

Choose a value on the interval $c > 7$ and see if this value makes the original inequality true.

$c = 10$

Step 11.3.2

Replace $c$ with $10$ in the original inequality.

$\frac{5}{2 (10) - 9} > 1$

Step 11.3.3

The left side $0.\overline{45}$ is not greater than the right side $1$ , which means that the given statement is false.

False

Step 11.4

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

$c < \frac{9}{2}$ False

$\frac{9}{2} < c < 7$ True

$c > 7$ False

$c < \frac{9}{2}$ False

$\frac{9}{2} < c < 7$ True

$c > 7$ False

Step 12

The solution consists of all of the true intervals.

$\frac{9}{2} < c < 7$

Step 13

The result can be shown in multiple forms.

Inequality Form:

$\frac{9}{2} < c < 7$

Interval Notation:

$(\frac{9}{2}, 7)$

Step 14