Algebra Examples

$\frac{5 x - 4}{2 x + 1} > 1$

Step 1

Subtract $1$ from both sides of the inequality.

$\frac{5 x - 4}{2 x + 1} - 1 > 0$

Step 2

Simplify $\frac{5 x - 4}{2 x + 1} - 1$ .

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Step 2.1

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

$\frac{5 x - 4}{2 x + 1} - 1 \cdot \frac{2 x + 1}{2 x + 1} > 0$

Step 2.2

Combine $- 1$ and $\frac{2 x + 1}{2 x + 1}$ .

$\frac{5 x - 4}{2 x + 1} + \frac{- (2 x + 1)}{2 x + 1} > 0$

Step 2.3

Combine the numerators over the common denominator.

$\frac{5 x - 4 - (2 x + 1)}{2 x + 1} > 0$

Step 2.4

Simplify the numerator.

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Step 2.4.1

Apply the distributive property.

$\frac{5 x - 4 - (2 x) - 1 \cdot 1}{2 x + 1} > 0$

Step 2.4.2

Multiply $2$ by $- 1$ .

$\frac{5 x - 4 - 2 x - 1 \cdot 1}{2 x + 1} > 0$

Step 2.4.3

Multiply $- 1$ by $1$ .

$\frac{5 x - 4 - 2 x - 1}{2 x + 1} > 0$

Step 2.4.4

Subtract $2 x$ from $5 x$ .

$\frac{3 x - 4 - 1}{2 x + 1} > 0$

Step 2.4.5

Subtract $1$ from $- 4$ .

$\frac{3 x - 5}{2 x + 1} > 0$

Step 3

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

$3 x - 5 = 0$

$2 x + 1 = 0$

Step 4

Add $5$ to both sides of the equation.

$3 x = 5$

Step 5

Divide each term in $3 x = 5$ by $3$ and simplify.

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Step 5.1

Divide each term in $3 x = 5$ by $3$ .

$\frac{3 x}{3} = \frac{5}{3}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $3$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{5}{3}$

Step 5.2.1.2

Divide $x$ by $1$ .

$x = \frac{5}{3}$

Step 6

Subtract $1$ from both sides of the equation.

$2 x = - 1$

Step 7

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

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Step 7.1

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

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

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $2$ .

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Step 7.2.1.1

Cancel the common factor.

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

Step 7.2.1.2

Divide $x$ by $1$ .

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

Step 7.3

Simplify the right side.

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Step 7.3.1

Move the negative in front of the fraction.

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

Step 8

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

$x = \frac{5}{3}$

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

Step 9

Consolidate the solutions.

$x = \frac{5}{3}, - \frac{1}{2}$

Step 10

Find the domain of $\frac{3 x - 5}{2 x + 1}$ .

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Step 10.1

Set the denominator in $\frac{3 x - 5}{2 x + 1}$ equal to $0$ to find where the expression is undefined.

$2 x + 1 = 0$

Step 10.2

Solve for $x$ .

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Step 10.2.1

Subtract $1$ from both sides of the equation.

$2 x = - 1$

Step 10.2.2

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

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Step 10.2.2.1

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

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

Step 10.2.2.2

Simplify the left side.

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Step 10.2.2.2.1

Cancel the common factor of $2$ .

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Step 10.2.2.2.1.1

Cancel the common factor.

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

Step 10.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 10.2.2.3

Simplify the right side.

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Step 10.2.2.3.1

Move the negative in front of the fraction.

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

Step 10.3

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

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

Step 11

Use each root to create test intervals.

$x < - \frac{1}{2}$

$- \frac{1}{2} < x < \frac{5}{3}$

$x > \frac{5}{3}$

Step 12

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 12.1

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

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Step 12.1.1

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

$x = - 3$

Step 12.1.2

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

$\frac{5 (- 3) - 4}{2 (- 3) + 1} > 1$

Step 12.1.3

The left side $3.8$ is greater than the right side $1$ , which means that the given statement is always true.

True

Step 12.2

Test a value on the interval $- \frac{1}{2} < x < \frac{5}{3}$ to see if it makes the inequality true.

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Step 12.2.1

Choose a value on the interval $- \frac{1}{2} < x < \frac{5}{3}$ and see if this value makes the original inequality true.

$x = 0$

Step 12.2.2

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

$\frac{5 (0) - 4}{2 (0) + 1} > 1$

Step 12.2.3

The left side $- 4$ is not greater than the right side $1$ , which means that the given statement is false.

False

Step 12.3

Test a value on the interval $x > \frac{5}{3}$ to see if it makes the inequality true.

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Step 12.3.1

Choose a value on the interval $x > \frac{5}{3}$ and see if this value makes the original inequality true.

$x = 4$

Step 12.3.2

Replace $x$ with $4$ in the original inequality.

$\frac{5 (4) - 4}{2 (4) + 1} > 1$

Step 12.3.3

The left side $1.\overline{7}$ is greater than the right side $1$ , which means that the given statement is always true.

True

Step 12.4

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

$x < - \frac{1}{2}$ True

$- \frac{1}{2} < x < \frac{5}{3}$ False

$x > \frac{5}{3}$ True

$x < - \frac{1}{2}$ True

$- \frac{1}{2} < x < \frac{5}{3}$ False

$x > \frac{5}{3}$ True

Step 13

The solution consists of all of the true intervals.

$x < - \frac{1}{2}$ or $x > \frac{5}{3}$

Step 14

The result can be shown in multiple forms.

Inequality Form:

$x < - \frac{1}{2} or x > \frac{5}{3}$

Interval Notation:

$(- \infty, - \frac{1}{2}) \cup (\frac{5}{3}, \infty)$

Step 15