Solve the Inequality for x 10/(7-x)>=x

$\frac{10}{7 - x} \geq x$

Step 1

Subtract $x$ from both sides of the inequality.

$\frac{10}{7 - x} - x \geq 0$

Step 2

Simplify $\frac{10}{7 - x} - x$ .

To write $- x$ as a fraction with a common denominator, multiply by $\frac{7 - x}{7 - x}$ .

$\frac{10}{7 - x} - x \cdot \frac{7 - x}{7 - x} \geq 0$

Combine $- x$ and $\frac{7 - x}{7 - x}$ .

$\frac{10}{7 - x} + \frac{- x (7 - x)}{7 - x} \geq 0$

Combine the numerators over the common denominator.

$\frac{10 - x (7 - x)}{7 - x} \geq 0$

Simplify the numerator.

Apply the distributive property.

$\frac{10 - x \cdot 7 - x (- x)}{7 - x} \geq 0$

Multiply $7$ by $- 1$ .

$\frac{10 - 7 x - x (- x)}{7 - x} \geq 0$

Rewrite using the commutative property of multiplication.

$\frac{10 - 7 x - 1 \cdot - 1 x \cdot x}{7 - x} \geq 0$

Simplify each term.

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$\frac{10 - 7 x - 1 \cdot - 1 (x \cdot x)}{7 - x} \geq 0$

Multiply $x$ by $x$ .

$\frac{10 - 7 x - 1 \cdot - 1 x^{2}}{7 - x} \geq 0$

Multiply $- 1$ by $- 1$ .

$\frac{10 - 7 x + 1 x^{2}}{7 - x} \geq 0$

Multiply $x^{2}$ by $1$ .

$\frac{10 - 7 x + x^{2}}{7 - x} \geq 0$

Factor $10 - 7 x + x^{2}$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $10$ and whose sum is $- 7$ .

$- 5, - 2$

Write the factored form using these integers.

$\frac{(x - 5) (x - 2)}{7 - x} \geq 0$

Step 3

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

$x - 5 = 0$

$x - 2 = 0$

$7 - x = 0$

Step 4

Add $5$ to both sides of the equation.

$x = 5$

Step 5

Add $2$ to both sides of the equation.

$x = 2$

Step 6

Subtract $7$ from both sides of the equation.

$- x = - 7$

Step 7

Divide each term in $- x = - 7$ by $- 1$ and simplify.

Divide each term in $- x = - 7$ by $- 1$ .

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

Simplify the left side.

Dividing two negative values results in a positive value.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Divide $- 7$ by $- 1$ .

$x = 7$

Step 8

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

$x = 5$

$x = 2$

$x = 7$

Step 9

Consolidate the solutions.

$x = 5, 2, 7$

Step 10

Find the domain of $\frac{(x - 5) (x - 2)}{7 - x}$ .

Set the denominator in $\frac{(x - 5) (x - 2)}{7 - x}$ equal to $0$ to find where the expression is undefined.

$7 - x = 0$

Solve for $x$ .

Subtract $7$ from both sides of the equation.

$- x = - 7$

Divide each term in $- x = - 7$ by $- 1$ and simplify.

Divide each term in $- x = - 7$ by $- 1$ .

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

Simplify the left side.

Dividing two negative values results in a positive value.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Divide $- 7$ by $- 1$ .

$x = 7$

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

$(- \infty, 7) \cup (7, \infty)$

Step 11

Use each root to create test intervals.

$x < 2$

$2 < x < 5$

$5 < x < 7$

$x > 7$

Step 12

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 $x < 2$ to see if it makes the inequality true.

Choose a value on the interval $x < 2$ and see if this value makes the original inequality true.

$x = 0$

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

$\frac{10}{7 - (0)} \geq 0$

The left side $1.\overline{428571}$ is greater than the right side $0$ , which means that the given statement is always true.

True

Test a value on the interval $2 < x < 5$ to see if it makes the inequality true.

Choose a value on the interval $2 < x < 5$ and see if this value makes the original inequality true.

$x = 4$

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

$\frac{10}{7 - (4)} \geq 4$

The left side $3.\overline{3}$ is less than the right side $4$ , which means that the given statement is false.

False

Test a value on the interval $5 < x < 7$ to see if it makes the inequality true.

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

$x = 6$

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

$\frac{10}{7 - (6)} \geq 6$

The left side $10$ is greater than the right side $6$ , which means that the given statement is always true.

True

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

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

$x = 10$

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

$\frac{10}{7 - (10)} \geq 10$

The left side $- 3.\overline{3}$ is less than the right side $10$ , which means that the given statement is false.

False

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

$x < 2$ True

$2 < x < 5$ False

$5 < x < 7$ True

$x > 7$ False

$x < 2$ True

$2 < x < 5$ False

$5 < x < 7$ True

$x > 7$ False

Step 13

The solution consists of all of the true intervals.

$x \leq 2$ or $5 \leq x < 7$

Step 14

The result can be shown in multiple forms.

Inequality Form:

$x \leq 2 or 5 \leq x < 7$

Interval Notation:

$(- \infty, 2] \cup [5, 7)$

Step 15

Solve the Inequality for x 10/(7-x)>=x

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