Algebra Examples

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

Step 1

Solve $0 < b \leq 1$ .

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Step 1.1

Multiply both sides by $b$ .

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

Step 1.2

Simplify.

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Step 1.2.1

Simplify the left side.

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Step 1.2.1.1

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

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Step 1.2.1.1.1

Apply the distributive property.

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

Step 1.2.1.1.2

Cancel the common factor of $b$ .

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Step 1.2.1.1.2.1

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

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

Step 1.2.1.1.2.2

Cancel the common factor.

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

Step 1.2.1.1.2.3

Rewrite the expression.

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

Step 1.2.2

Simplify the right side.

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Step 1.2.2.1

Cancel the common factor of $b$ .

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Step 1.2.2.1.1

Cancel the common factor.

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

Step 1.2.2.1.2

Rewrite the expression.

$7 b - 2 = 5$

Step 1.3

Solve for $b$ .

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Step 1.3.1

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

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Step 1.3.1.1

Add $2$ to both sides of the equation.

$7 b = 5 + 2$

Step 1.3.1.2

Add $5$ and $2$ .

$7 b = 7$

Step 1.3.2

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

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Step 1.3.2.1

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

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

Step 1.3.2.2

Simplify the left side.

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Step 1.3.2.2.1

Cancel the common factor of $7$ .

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Step 1.3.2.2.1.1

Cancel the common factor.

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

Step 1.3.2.2.1.2

Divide $b$ by $1$ .

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

Step 1.3.2.3

Simplify the right side.

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Step 1.3.2.3.1

Divide $7$ by $7$ .

$b = 1$

Step 1.4

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

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Step 1.4.1

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

$b = 0$

Step 1.4.2

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

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

Step 1.5

Use each root to create test intervals.

$b < 0$

$0 < b < 1$

$b > 1$

Step 1.6

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 1.6.1

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

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Step 1.6.1.1

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

$b = - 2$

Step 1.6.1.2

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

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

Step 1.6.1.3

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

False

Step 1.6.2

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

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Step 1.6.2.1

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

$b = 0.5$

Step 1.6.2.2

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

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

Step 1.6.2.3

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

True

Step 1.6.3

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

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Step 1.6.3.1

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

$b = 4$

Step 1.6.3.2

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

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

Step 1.6.3.3

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

False

Step 1.6.4

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

Step 1.7

The solution consists of all of the true intervals.

$0 < b \leq 1$

Step 2

Use the inequality $0 < b \leq 1$ to build the set notation.

${b | 0 < b \leq 1}$

Step 3