Algebra Examples

$\frac{4}{x} - 3 > \frac{2}{x} - 7$

Step 1

Move all terms containing variables to the left side of the inequality.

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Step 1.1

Subtract $\frac{2}{x}$ from both sides of the inequality.

$\frac{4}{x} - 3 - \frac{2}{x} > - 7$

Step 1.2

Combine the numerators over the common denominator.

$- 3 + \frac{4 - 2}{x} > - 7$

Step 1.3

Subtract $2$ from $4$ .

$- 3 + \frac{2}{x} > - 7$

Step 2

Move all terms not containing $x$ to the right side of the inequality.

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Step 2.1

Add $3$ to both sides of the inequality.

$\frac{2}{x} > - 7 + 3$

Step 2.2

Add $- 7$ and $3$ .

$\frac{2}{x} > - 4$

Step 3

Multiply both sides by $x$ .

$\frac{2}{x} x = - 4 x$

Step 4

Simplify the left side.

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Step 4.1

Cancel the common factor of $x$ .

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Step 4.1.1

Cancel the common factor.

$\frac{2}{x} x = - 4 x$

Step 4.1.2

Rewrite the expression.

$2 = - 4 x$

Step 5

Solve for $x$ .

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Step 5.1

Rewrite the equation as $- 4 x = 2$ .

$- 4 x = 2$

Step 5.2

Divide each term in $- 4 x = 2$ by $- 4$ and simplify.

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Step 5.2.1

Divide each term in $- 4 x = 2$ by $- 4$ .

$\frac{- 4 x}{- 4} = \frac{2}{- 4}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $- 4$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{- 4 x}{- 4} = \frac{2}{- 4}$

Step 5.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{2}{- 4}$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Cancel the common factor of $2$ and $- 4$ .

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Step 5.2.3.1.1

Factor $2$ out of $2$ .

$x = \frac{2 (1)}{- 4}$

Step 5.2.3.1.2

Cancel the common factors.

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Step 5.2.3.1.2.1

Factor $2$ out of $- 4$ .

$x = \frac{2 \cdot 1}{2 \cdot - 2}$

Step 5.2.3.1.2.2

Cancel the common factor.

$x = \frac{2 \cdot 1}{2 \cdot - 2}$

Step 5.2.3.1.2.3

Rewrite the expression.

$x = \frac{1}{- 2}$

Step 5.2.3.2

Move the negative in front of the fraction.

$x = - \frac{1}{2}$

Step 6

Find the domain of $4 + \frac{2}{x}$ .

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Step 6.1

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

$x = 0$

Step 6.2

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

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

Step 7

Use each root to create test intervals.

$x < - \frac{1}{2}$

$- \frac{1}{2} < x < 0$

$x > 0$

Step 8

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 8.1

Test a value on the interval $x < - \frac{1}{2}$ to see if it makes the inequality true.

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Step 8.1.1

Choose a value on the interval $x < - \frac{1}{2}$ and see if this value makes the original inequality true.

$x = - 3$

Step 8.1.2

Replace $x$ with $- 3$ in the original inequality.

$\frac{4}{- 3} - 3 > \frac{2}{- 3} - 7$

Step 8.1.3

The left side $- 4.\overline{3}$ is greater than the right side $- 7.\overline{6}$ , which means that the given statement is always true.

True

Step 8.2

Test a value on the interval $- \frac{1}{2} < x < 0$ to see if it makes the inequality true.

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Step 8.2.1

Choose a value on the interval $- \frac{1}{2} < x < 0$ and see if this value makes the original inequality true.

$x = - 0.25$

Step 8.2.2

Replace $x$ with $- 0.25$ in the original inequality.

$\frac{4}{- 0.25} - 3 > \frac{2}{- 0.25} - 7$

Step 8.2.3

The left side $- 19$ is not greater than the right side $- 15$ , which means that the given statement is false.

False

Step 8.3

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

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Step 8.3.1

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

$x = 2$

Step 8.3.2

Replace $x$ with $2$ in the original inequality.

$\frac{4}{2} - 3 > \frac{2}{2} - 7$

Step 8.3.3

The left side $- 1$ is greater than the right side $- 6$ , which means that the given statement is always true.

True

Step 8.4

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

$x < - \frac{1}{2}$ True

$- \frac{1}{2} < x < 0$ False

$x > 0$ True

$x < - \frac{1}{2}$ True

$- \frac{1}{2} < x < 0$ False

$x > 0$ True

Step 9

The solution consists of all of the true intervals.

$x < - \frac{1}{2}$ or $x > 0$

Step 10

The result can be shown in multiple forms.

Inequality Form:

$x < - \frac{1}{2} or x > 0$

Interval Notation:

$(- \infty, - \frac{1}{2}) \cup (0, \infty)$

Step 11