Algebra Examples

$18 (q - \frac{3}{4}) \leq - 2 (q + \frac{7}{4})$

Step 1

Simplify $18 (q - \frac{3}{4})$ .

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Step 1.1

Rewrite.

$0 + 0 + 18 (q - \frac{3}{4}) \leq - 2 (q + \frac{7}{4})$

Step 1.2

Simplify by adding zeros.

$18 (q - \frac{3}{4}) \leq - 2 (q + \frac{7}{4})$

Step 1.3

Apply the distributive property.

$18 q + 18 (- \frac{3}{4}) \leq - 2 (q + \frac{7}{4})$

Step 1.4

Cancel the common factor of $2$ .

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Step 1.4.1

Move the leading negative in $- \frac{3}{4}$ into the numerator.

$18 q + 18 (\frac{- 3}{4}) \leq - 2 (q + \frac{7}{4})$

Step 1.4.2

Factor $2$ out of $18$ .

$18 q + 2 (9) \frac{- 3}{4} \leq - 2 (q + \frac{7}{4})$

Step 1.4.3

Factor $2$ out of $4$ .

$18 q + 2 \cdot 9 \frac{- 3}{2 \cdot 2} \leq - 2 (q + \frac{7}{4})$

Step 1.4.4

Cancel the common factor.

$18 q + 2 \cdot 9 \frac{- 3}{2 \cdot 2} \leq - 2 (q + \frac{7}{4})$

Step 1.4.5

Rewrite the expression.

$18 q + 9 (\frac{- 3}{2}) \leq - 2 (q + \frac{7}{4})$

Step 1.5

Combine $9$ and $\frac{- 3}{2}$ .

$18 q + \frac{9 \cdot - 3}{2} \leq - 2 (q + \frac{7}{4})$

Step 1.6

Simplify the expression.

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Step 1.6.1

Multiply $9$ by $- 3$ .

$18 q + \frac{- 27}{2} \leq - 2 (q + \frac{7}{4})$

Step 1.6.2

Move the negative in front of the fraction.

$18 q - \frac{27}{2} \leq - 2 (q + \frac{7}{4})$

Step 2

Simplify $- 2 (q + \frac{7}{4})$ .

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Step 2.1

Simplify terms.

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Step 2.1.1

Apply the distributive property.

$18 q - \frac{27}{2} \leq - 2 q - 2 (\frac{7}{4})$

Step 2.1.2

Cancel the common factor of $2$ .

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Step 2.1.2.1

Factor $2$ out of $- 2$ .

$18 q - \frac{27}{2} \leq - 2 q + 2 (- 1) \frac{7}{4}$

Step 2.1.2.2

Factor $2$ out of $4$ .

$18 q - \frac{27}{2} \leq - 2 q + 2 \cdot - 1 \frac{7}{2 \cdot 2}$

Step 2.1.2.3

Cancel the common factor.

$18 q - \frac{27}{2} \leq - 2 q + 2 \cdot - 1 \frac{7}{2 \cdot 2}$

Step 2.1.2.4

Rewrite the expression.

$18 q - \frac{27}{2} \leq - 2 q - 1 (\frac{7}{2})$

Step 2.2

Rewrite $- 1 (\frac{7}{2})$ as $- (\frac{7}{2})$ .

$18 q - \frac{27}{2} \leq - 2 q - \frac{7}{2}$

Step 3

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

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Step 3.1

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

$18 q - \frac{27}{2} + 2 q \leq - \frac{7}{2}$

Step 3.2

Add $18 q$ and $2 q$ .

$20 q - \frac{27}{2} \leq - \frac{7}{2}$

Step 4

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

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Step 4.1

Add $\frac{27}{2}$ to both sides of the inequality.

$20 q \leq - \frac{7}{2} + \frac{27}{2}$

Step 4.2

Combine the numerators over the common denominator.

$20 q \leq \frac{- 7 + 27}{2}$

Step 4.3

Add $- 7$ and $27$ .

$20 q \leq \frac{20}{2}$

Step 4.4

Divide $20$ by $2$ .

$20 q \leq 10$

Step 5

Divide each term in $20 q \leq 10$ by $20$ and simplify.

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Step 5.1

Divide each term in $20 q \leq 10$ by $20$ .

$\frac{20 q}{20} \leq \frac{10}{20}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $20$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{20 q}{20} \leq \frac{10}{20}$

Step 5.2.1.2

Divide $q$ by $1$ .

$q \leq \frac{10}{20}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Cancel the common factor of $10$ and $20$ .

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Step 5.3.1.1

Factor $10$ out of $10$ .

$q \leq \frac{10 (1)}{20}$

Step 5.3.1.2

Cancel the common factors.

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Step 5.3.1.2.1

Factor $10$ out of $20$ .

$q \leq \frac{10 \cdot 1}{10 \cdot 2}$

Step 5.3.1.2.2

Cancel the common factor.

$q \leq \frac{10 \cdot 1}{10 \cdot 2}$

Step 5.3.1.2.3

Rewrite the expression.

$q \leq \frac{1}{2}$

Step 6

The result can be shown in multiple forms.

Inequality Form:

$q \leq \frac{1}{2}$

Interval Notation:

$(- \infty, \frac{1}{2}]$