Algebra Examples

$5 - \frac{3}{a} < \frac{7}{a}$

Step 1

Multiply both sides by $a$ .

$(5 - \frac{3}{a}) a = \frac{7}{a} a$

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Simplify $(5 - \frac{3}{a}) a$ .

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Step 2.1.1.1

Apply the distributive property.

$5 a - \frac{3}{a} a = \frac{7}{a} a$

Step 2.1.1.2

Cancel the common factor of $a$ .

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Step 2.1.1.2.1

Move the leading negative in $- \frac{3}{a}$ into the numerator.

$5 a + \frac{- 3}{a} a = \frac{7}{a} a$

Step 2.1.1.2.2

Cancel the common factor.

$5 a + \frac{- 3}{a} a = \frac{7}{a} a$

Step 2.1.1.2.3

Rewrite the expression.

$5 a - 3 = \frac{7}{a} a$

Step 2.2

Simplify the right side.

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Step 2.2.1

Cancel the common factor of $a$ .

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Step 2.2.1.1

Cancel the common factor.

$5 a - 3 = \frac{7}{a} a$

Step 2.2.1.2

Rewrite the expression.

$5 a - 3 = 7$

Step 3

Solve for $a$ .

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Step 3.1

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

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Step 3.1.1

Add $3$ to both sides of the equation.

$5 a = 7 + 3$

Step 3.1.2

Add $7$ and $3$ .

$5 a = 10$

Step 3.2

Divide each term in $5 a = 10$ by $5$ and simplify.

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Step 3.2.1

Divide each term in $5 a = 10$ by $5$ .

$\frac{5 a}{5} = \frac{10}{5}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $5$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{5 a}{5} = \frac{10}{5}$

Step 3.2.2.1.2

Divide $a$ by $1$ .

$a = \frac{10}{5}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Divide $10$ by $5$ .

$a = 2$

Step 4

Find the domain of $5 - \frac{10}{a}$ .

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Step 4.1

Set the denominator in $\frac{10}{a}$ equal to $0$ to find where the expression is undefined.

$a = 0$

Step 4.2

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

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

Step 5

Use each root to create test intervals.

$a < 0$

$0 < a < 2$

$a > 2$

Step 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 6.1

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

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Step 6.1.1

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

$a = - 2$

Step 6.1.2

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

$5 - \frac{3}{- 2} < \frac{7}{- 2}$

Step 6.1.3

The left side $6.5$ is not less than the right side $- 3.5$ , which means that the given statement is false.

False

Step 6.2

Test a value on the interval $0 < a < 2$ to see if it makes the inequality true.

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Step 6.2.1

Choose a value on the interval $0 < a < 2$ and see if this value makes the original inequality true.

$a = 1$

Step 6.2.2

Replace $a$ with $1$ in the original inequality.

$5 - \frac{3}{1} < \frac{7}{1}$

Step 6.2.3

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

True

Step 6.3

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

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Step 6.3.1

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

$a = 4$

Step 6.3.2

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

$5 - \frac{3}{4} < \frac{7}{4}$

Step 6.3.3

The left side $4.25$ is not less than the right side $1.75$ , which means that the given statement is false.

False

Step 6.4

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

$a < 0$ False

$0 < a < 2$ True

$a > 2$ False

$a < 0$ False

$0 < a < 2$ True

$a > 2$ False

Step 7

The solution consists of all of the true intervals.

$0 < a < 2$

Step 8

The result can be shown in multiple forms.

Inequality Form:

$0 < a < 2$

Interval Notation:

$(0, 2)$

Step 9