Algebra Examples

$\frac{6 x + 5}{x - 4} > \frac{8 - 2 x}{x - 4}$

Step 1

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$6 x + 5 > 8 - 2 x$

Step 2

Move all terms containing $x$ to the left side of the inequality.

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Step 2.1

Add $2 x$ to both sides of the inequality.

$6 x + 5 + 2 x > 8$

Step 2.2

Add $6 x$ and $2 x$ .

$8 x + 5 > 8$

Step 3

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

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Step 3.1

Subtract $5$ from both sides of the inequality.

$8 x > 8 - 5$

Step 3.2

Subtract $5$ from $8$ .

$8 x > 3$

Step 4

Divide each term in $8 x > 3$ by $8$ and simplify.

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Step 4.1

Divide each term in $8 x > 3$ by $8$ .

$\frac{8 x}{8} > \frac{3}{8}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $8$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{8 x}{8} > \frac{3}{8}$

Step 4.2.1.2

Divide $x$ by $1$ .

$x > \frac{3}{8}$

Step 5

Find the domain of $\frac{8 x - 3}{x - 4}$ .

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Step 5.1

Set the denominator in $\frac{8 x - 3}{x - 4}$ equal to $0$ to find where the expression is undefined.

$x - 4 = 0$

Step 5.2

Add $4$ to both sides of the equation.

$x = 4$

Step 5.3

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

$(- \infty, 4) \cup (4, \infty)$

Step 6

Use each root to create test intervals.

$x < \frac{3}{8}$

$\frac{3}{8} < x < 4$

$x > 4$

Step 7

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 7.1

Test a value on the interval $x < \frac{3}{8}$ to see if it makes the inequality true.

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Step 7.1.1

Choose a value on the interval $x < \frac{3}{8}$ and see if this value makes the original inequality true.

$x = 0$

Step 7.1.2

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

$\frac{6 (0) + 5}{(0) - 4} > \frac{8 - 2 \cdot 0}{(0) - 4}$

Step 7.1.3

The left side $- 1.25$ is greater than the right side $- 2$ , which means that the given statement is always true.

True

Step 7.2

Test a value on the interval $\frac{3}{8} < x < 4$ to see if it makes the inequality true.

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Step 7.2.1

Choose a value on the interval $\frac{3}{8} < x < 4$ and see if this value makes the original inequality true.

$x = 2$

Step 7.2.2

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

$\frac{6 (2) + 5}{(2) - 4} > \frac{8 - 2 \cdot 2}{(2) - 4}$

Step 7.2.3

The left side $- 8.5$ is not greater than the right side $- 2$ , which means that the given statement is false.

False

Step 7.3

Test a value on the interval $x > 4$ to see if it makes the inequality true.

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Step 7.3.1

Choose a value on the interval $x > 4$ and see if this value makes the original inequality true.

$x = 6$

Step 7.3.2

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

$\frac{6 (6) + 5}{(6) - 4} > \frac{8 - 2 \cdot 6}{(6) - 4}$

Step 7.3.3

The left side $20.5$ is greater than the right side $- 2$ , which means that the given statement is always true.

True

Step 7.4

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

$x < \frac{3}{8}$ True

$\frac{3}{8} < x < 4$ False

$x > 4$ True

$x < \frac{3}{8}$ True

$\frac{3}{8} < x < 4$ False

$x > 4$ True

Step 8

The solution consists of all of the true intervals.

$x < \frac{3}{8}$ or $x > 4$

Step 9

The result can be shown in multiple forms.

Inequality Form:

$x < \frac{3}{8} or x > 4$

Interval Notation:

$(- \infty, \frac{3}{8}) \cup (4, \infty)$

Step 10