Pre-Algebra Examples

$\frac{1}{4 c} < 8$

Step 1

Multiply both sides by $4 c$ .

$\frac{1}{4 c} (4 c) = 8 (4 c)$

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Simplify $\frac{1}{4 c} (4 c)$ .

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Step 2.1.1.1

Rewrite using the commutative property of multiplication.

$4 \frac{1}{4 c} c = 8 (4 c)$

Step 2.1.1.2

Cancel the common factor of $4$ .

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Step 2.1.1.2.1

Factor $4$ out of $4 c$ .

$4 \frac{1}{4 (c)} c = 8 (4 c)$

Step 2.1.1.2.2

Cancel the common factor.

$4 \frac{1}{4 c} c = 8 (4 c)$

Step 2.1.1.2.3

Rewrite the expression.

$\frac{1}{c} c = 8 (4 c)$

Step 2.1.1.3

Cancel the common factor of $c$ .

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Step 2.1.1.3.1

Cancel the common factor.

$\frac{1}{c} c = 8 (4 c)$

Step 2.1.1.3.2

Rewrite the expression.

$1 = 8 (4 c)$

Step 2.2

Simplify the right side.

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Step 2.2.1

Multiply $4$ by $8$ .

$1 = 32 c$

Step 3

Solve for $c$ .

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Step 3.1

Rewrite the equation as $32 c = 1$ .

$32 c = 1$

Step 3.2

Divide each term in $32 c = 1$ by $32$ and simplify.

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Step 3.2.1

Divide each term in $32 c = 1$ by $32$ .

$\frac{32 c}{32} = \frac{1}{32}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $32$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{32 c}{32} = \frac{1}{32}$

Step 3.2.2.1.2

Divide $c$ by $1$ .

$c = \frac{1}{32}$

Step 4

Find the domain of $\frac{1}{4 c} - 8$ .

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Step 4.1

Set the denominator in $\frac{1}{4 c}$ equal to $0$ to find where the expression is undefined.

$4 c = 0$

Step 4.2

Divide each term in $4 c = 0$ by $4$ and simplify.

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Step 4.2.1

Divide each term in $4 c = 0$ by $4$ .

$\frac{4 c}{4} = \frac{0}{4}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $4$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{4 c}{4} = \frac{0}{4}$

Step 4.2.2.1.2

Divide $c$ by $1$ .

$c = \frac{0}{4}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Divide $0$ by $4$ .

$c = 0$

Step 4.3

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

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

Step 5

Use each root to create test intervals.

$c < 0$

$0 < c < \frac{1}{32}$

$c > \frac{1}{32}$

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

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Step 6.1.1

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

$c = - 2$

Step 6.1.2

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

$\frac{1}{4 (- 2)} < 8$

Step 6.1.3

The left side $- 0.125$ is less than the right side $8$ , which means that the given statement is always true.

True

Step 6.2

Test a value on the interval $0 < c < \frac{1}{32}$ to see if it makes the inequality true.

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Step 6.2.1

Choose a value on the interval $0 < c < \frac{1}{32}$ and see if this value makes the original inequality true.

$c = 0.02$

Step 6.2.2

Replace $c$ with $0.02$ in the original inequality.

$\frac{1}{4 (0.02)} < 8$

Step 6.2.3

The left side $12.5$ is not less than the right side $8$ , which means that the given statement is false.

False

Step 6.3

Test a value on the interval $c > \frac{1}{32}$ to see if it makes the inequality true.

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Step 6.3.1

Choose a value on the interval $c > \frac{1}{32}$ and see if this value makes the original inequality true.

$c = 3$

Step 6.3.2

Replace $c$ with $3$ in the original inequality.

$\frac{1}{4 (3)} < 8$

Step 6.3.3

The left side $0.08\overline{3}$ is less than the right side $8$ , which means that the given statement is always true.

True

Step 6.4

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

$c < 0$ True

$0 < c < \frac{1}{32}$ False

$c > \frac{1}{32}$ True

$c < 0$ True

$0 < c < \frac{1}{32}$ False

$c > \frac{1}{32}$ True

Step 7

The solution consists of all of the true intervals.

$c < 0$ or $c > \frac{1}{32}$

Step 8

The result can be shown in multiple forms.

Inequality Form:

$c < 0 or c > \frac{1}{32}$

Interval Notation:

$(- \infty, 0) \cup (\frac{1}{32}, \infty)$

Step 9