Solve for x (2x+7)/(x-1)<=1

$\frac{2 x + 7}{x - 1} \leq 1$

Step 1

Subtract $1$ from both sides of the inequality.

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

Step 2

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

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Apply the distributive property.

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

Multiply $- 1$ by $- 1$ .

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

Subtract $x$ from $2 x$ .

$\frac{x + 7 + 1}{x - 1} \leq 0$

Add $7$ and $1$ .

$\frac{x + 8}{x - 1} \leq 0$

Step 3

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

$x + 8 = 0$

$x - 1 = 0$

Step 4

Subtract $8$ from both sides of the equation.

$x = - 8$

Step 5

Add $1$ to both sides of the equation.

$x = 1$

Step 6

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

$x = - 8$

$x = 1$

Step 7

Consolidate the solutions.

$x = - 8, 1$

Step 8

Find the domain of $\frac{x + 8}{x - 1}$ .

Set the denominator in $\frac{x + 8}{x - 1}$ equal to $0$ to find where the expression is undefined.

$x - 1 = 0$

Add $1$ to both sides of the equation.

$x = 1$

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

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

Step 9

Use each root to create test intervals.

$x < - 8$

$- 8 < x < 1$

$x > 1$

Step 10

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

Test a value on the interval $x < - 8$ to see if it makes the inequality true.

Choose a value on the interval $x < - 8$ and see if this value makes the original inequality true.

$x = - 10$

Replace $x$ with $- 10$ in the original inequality.

$\frac{2 (- 10) + 7}{(- 10) - 1} \leq 1$

The left side $1.\overline{18}$ is greater than the right side $1$ , which means that the given statement is false.

False

Test a value on the interval $- 8 < x < 1$ to see if it makes the inequality true.

Choose a value on the interval $- 8 < x < 1$ and see if this value makes the original inequality true.

$x = 0$

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

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

The left side $- 7$ is less than the right side $1$ , which means that the given statement is always true.

True

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

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

$x = 4$

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

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

The left side $5$ is greater than the right side $1$ , which means that the given statement is false.

False

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

$x < - 8$ False

$- 8 < x < 1$ True

$x > 1$ False

$x < - 8$ False

$- 8 < x < 1$ True

$x > 1$ False

Step 11

The solution consists of all of the true intervals.

$- 8 \leq x < 1$

Step 12

The result can be shown in multiple forms.

Inequality Form:

$- 8 \leq x < 1$

Interval Notation:

$[- 8, 1)$

Step 13

Solve for x (2x+7)/(x-1)<=1

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