Algebra Examples

$\frac{5 x + 7}{x - 1} < - 2$

Step 1

Add $2$ to both sides of the inequality.

$\frac{5 x + 7}{x - 1} + 2 < 0$

Step 2

Simplify $\frac{5 x + 7}{x - 1} + 2$ .

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Step 2.1

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

$\frac{5 x + 7}{x - 1} + \frac{2 (x - 1)}{x - 1} < 0$

Step 2.2

Combine the numerators over the common denominator.

$\frac{5 x + 7 + 2 (x - 1)}{x - 1} < 0$

Step 2.3

Simplify the numerator.

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Step 2.3.1

Apply the distributive property.

$\frac{5 x + 7 + 2 x + 2 \cdot - 1}{x - 1} < 0$

Step 2.3.2

Multiply $2$ by $- 1$ .

$\frac{5 x + 7 + 2 x - 2}{x - 1} < 0$

Step 2.3.3

Add $5 x$ and $2 x$ .

$\frac{7 x + 7 - 2}{x - 1} < 0$

Step 2.3.4

Subtract $2$ from $7$ .

$\frac{7 x + 5}{x - 1} < 0$

Step 3

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

$7 x + 5 = 0$

$x - 1 = 0$

Step 4

Subtract $5$ from both sides of the equation.

$7 x = - 5$

Step 5

Divide each term in $7 x = - 5$ by $7$ and simplify.

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Step 5.1

Divide each term in $7 x = - 5$ by $7$ .

$\frac{7 x}{7} = \frac{- 5}{7}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $7$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{7 x}{7} = \frac{- 5}{7}$

Step 5.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 5}{7}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Move the negative in front of the fraction.

$x = - \frac{5}{7}$

Step 6

Add $1$ to both sides of the equation.

$x = 1$

Step 7

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

$x = - \frac{5}{7}$

$x = 1$

Step 8

Consolidate the solutions.

$x = - \frac{5}{7}, 1$

Step 9

Find the domain of $\frac{7 x + 5}{x - 1}$ .

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Step 9.1

Set the denominator in $\frac{7 x + 5}{x - 1}$ equal to $0$ to find where the expression is undefined.

$x - 1 = 0$

Step 9.2

Add $1$ to both sides of the equation.

$x = 1$

Step 9.3

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

$(- \infty, 1) \cup (1, \infty)$

Step 10

Use each root to create test intervals.

$x < - \frac{5}{7}$

$- \frac{5}{7} < x < 1$

$x > 1$

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 $x < - \frac{5}{7}$ to see if it makes the inequality true.

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Step 11.1.1

Choose a value on the interval $x < - \frac{5}{7}$ and see if this value makes the original inequality true.

$x = - 3$

Step 11.1.2

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

$\frac{5 (- 3) + 7}{(- 3) - 1} < - 2$

Step 11.1.3

The left side $2$ is not less than the right side $- 2$ , which means that the given statement is false.

False

Step 11.2

Test a value on the interval $- \frac{5}{7} < x < 1$ to see if it makes the inequality true.

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Step 11.2.1

Choose a value on the interval $- \frac{5}{7} < x < 1$ and see if this value makes the original inequality true.

$x = 0$

Step 11.2.2

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

$\frac{5 (0) + 7}{(0) - 1} < - 2$

Step 11.2.3

The left side $- 7$ is less than the right side $- 2$ , which means that the given statement is always true.

True

Step 11.3

Test a value on the interval $x > 1$ to see if it makes the inequality true.

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Step 11.3.1

Choose a value on the interval $x > 1$ and see if this value makes the original inequality true.

$x = 4$

Step 11.3.2

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

$\frac{5 (4) + 7}{(4) - 1} < - 2$

Step 11.3.3

The left side $9$ is not less than the right side $- 2$ , which means that the given statement is false.

False

Step 11.4

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

$x < - \frac{5}{7}$ False

$- \frac{5}{7} < x < 1$ True

$x > 1$ False

$x < - \frac{5}{7}$ False

$- \frac{5}{7} < x < 1$ True

$x > 1$ False

Step 12

The solution consists of all of the true intervals.

$- \frac{5}{7} < x < 1$

Step 13

Convert the inequality to interval notation.

$(- \frac{5}{7}, 1)$

Step 14