Solve the Inequality for b 7-2/b<=5/b

$7 - \frac{2}{b} \leq \frac{5}{b}$

Multiply both sides by $b$ .

$(7 - \frac{2}{b}) b = \frac{5}{b} b$

Simplify.

Simplify the left side.

Simplify $(7 - \frac{2}{b}) b$ .

Apply the distributive property.

$7 b - \frac{2}{b} b = \frac{5}{b} b$

Cancel the common factor of $b$ .

Move the leading negative in $- \frac{2}{b}$ into the numerator.

$7 b + \frac{- 2}{b} b = \frac{5}{b} b$

Cancel the common factor.

$7 b + \frac{- 2}{b} b = \frac{5}{b} b$

Rewrite the expression.

$7 b - 2 = \frac{5}{b} b$

Simplify the right side.

Cancel the common factor of $b$ .

Cancel the common factor.

$7 b - 2 = \frac{5}{b} b$

Rewrite the expression.

$7 b - 2 = 5$

Solve for $b$ .

Move all terms not containing $b$ to the right side of the equation.

Add $2$ to both sides of the equation.

$7 b = 5 + 2$

Add $5$ and $2$ .

$7 b = 7$

Divide each term in $7 b = 7$ by $7$ and simplify.

Divide each term in $7 b = 7$ by $7$ .

$\frac{7 b}{7} = \frac{7}{7}$

Simplify the left side.

Cancel the common factor of $7$ .

Cancel the common factor.

$\frac{7 b}{7} = \frac{7}{7}$

Divide $b$ by $1$ .

$b = \frac{7}{7}$

Simplify the right side.

Divide $7$ by $7$ .

$b = 1$

Find the domain of $7 - \frac{7}{b}$ .

Set the denominator in $\frac{7}{b}$ equal to $0$ to find where the expression is undefined.

$b = 0$

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

$(- \infty, 0) \cup (0, \infty)$

Use each root to create test intervals.

$b < 0$

$0 < b < 1$

$b > 1$

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 $b < 0$ to see if it makes the inequality true.

Choose a value on the interval $b < 0$ and see if this value makes the original inequality true.

$b = - 2$

Replace $b$ with $- 2$ in the original inequality.

$7 - \frac{2}{- 2} \leq \frac{5}{- 2}$

The left side $8$ is greater than the right side $- 2.5$ , which means that the given statement is false.

False

Test a value on the interval $0 < b < 1$ to see if it makes the inequality true.

Choose a value on the interval $0 < b < 1$ and see if this value makes the original inequality true.

$b = 0.5$

Replace $b$ with $0.5$ in the original inequality.

$7 - \frac{2}{0.5} \leq \frac{5}{0.5}$

The left side $3$ is less than the right side $10$ , which means that the given statement is always true.

True

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

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

$b = 4$

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

$7 - \frac{2}{4} \leq \frac{5}{4}$

The left side $6.5$ is greater than the right side $1.25$ , which means that the given statement is false.

False

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

$b < 0$ False

$0 < b < 1$ True

$b > 1$ False

$b < 0$ False

$0 < b < 1$ True

$b > 1$ False

The solution consists of all of the true intervals.

$0 < b \leq 1$

The result can be shown in multiple forms.

Inequality Form:

$0 < b \leq 1$

Interval Notation:

$(0, 1]$

Solve the Inequality for b 7-2/b<=5/b

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