Algebra Examples

$30 < 5 x - 5 \frac{x - 6}{2} \leq 60$

Step 1

Simplify each term.

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Step 1.1

Combine $- 5$ and $\frac{x - 6}{2}$ .

$30 < 5 x + \frac{- 5 (x - 6)}{2} \leq 60$

Step 1.2

Move the negative in front of the fraction.

$30 < 5 x - \frac{5 (x - 6)}{2} \leq 60$

Step 2

To write $5 x$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$30 < 5 x \cdot \frac{2}{2} - \frac{5 (x - 6)}{2} \leq 60$

Step 3

Simplify terms.

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Step 3.1

Combine $5 x$ and $\frac{2}{2}$ .

$30 < \frac{5 x \cdot 2}{2} - \frac{5 (x - 6)}{2} \leq 60$

Step 3.2

Combine the numerators over the common denominator.

$30 < \frac{5 x \cdot 2 - 5 (x - 6)}{2} \leq 60$

Step 4

Simplify the numerator.

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Step 4.1

Factor $5$ out of $5 x \cdot 2 - 5 (x - 6)$ .

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Step 4.1.1

Factor $5$ out of $5 x \cdot 2$ .

$30 < \frac{5 (x \cdot 2) - 5 (x - 6)}{2} \leq 60$

Step 4.1.2

Factor $5$ out of $- 5 (x - 6)$ .

$30 < \frac{5 (x \cdot 2) + 5 (- (x - 6))}{2} \leq 60$

Step 4.1.3

Factor $5$ out of $5 (x \cdot 2) + 5 (- (x - 6))$ .

$30 < \frac{5 (x \cdot 2 - (x - 6))}{2} \leq 60$

Step 4.2

Move $2$ to the left of $x$ .

$30 < \frac{5 (2 \cdot x - (x - 6))}{2} \leq 60$

Step 4.3

Apply the distributive property.

$30 < \frac{5 (2 x - x + 6)}{2} \leq 60$

Step 4.4

Multiply $- 1$ by $- 6$ .

$30 < \frac{5 (2 x - x + 6)}{2} \leq 60$

Step 4.5

Subtract $x$ from $2 x$ .

$30 < \frac{5 (x + 6)}{2} \leq 60$

Step 5

Multiply each term in the inequality by $2$ .

$30 \cdot 2 < \frac{5 (x + 6)}{2} \cdot 2 \leq 60 \cdot 2$

Step 6

Multiply $30$ by $2$ .

$60 < \frac{5 (x + 6)}{2} \cdot 2 \leq 60 \cdot 2$

Step 7

Cancel the common factor of $2$ .

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Step 7.1

Cancel the common factor.

$60 < \frac{5 (x + 6)}{2} \cdot 2 \leq 60 \cdot 2$

Step 7.2

Rewrite the expression.

$60 < 5 (x + 6) \leq 60 \cdot 2$

Step 8

Apply the distributive property.

$60 < 5 x + 5 \cdot 6 \leq 60 \cdot 2$

Step 9

Multiply $5$ by $6$ .

$60 < 5 x + 30 \leq 60 \cdot 2$

Step 10

Multiply $60$ by $2$ .

$60 < 5 x + 30 \leq 120$

Step 11

Move all terms not containing $x$ from the center section of the inequality.

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Step 11.1

Subtract $30$ from each section of the inequality because it does not contain the variable we are trying to solve for.

$60 - 30 < 5 x \leq 120 - 30$

Step 11.2

Subtract $30$ from $60$ .

$30 < 5 x \leq 120 - 30$

Step 11.3

Subtract $30$ from $120$ .

$30 < 5 x \leq 90$

Step 12

Divide each term in the inequality by $5$ .

$\frac{30}{5} < \frac{5 x}{5} \leq \frac{90}{5}$

Step 13

Divide $30$ by $5$ .

$6 < \frac{5 x}{5} \leq \frac{90}{5}$

Step 14

Cancel the common factor of $5$ .

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Step 14.1

Cancel the common factor.

$6 < \frac{5 x}{5} \leq \frac{90}{5}$

Step 14.2

Divide $x$ by $1$ .

$6 < x \leq \frac{90}{5}$

Step 15

Divide $90$ by $5$ .

$6 < x \leq 18$

Step 16

The result can be shown in multiple forms.

Inequality Form:

$6 < x \leq 18$

Interval Notation:

$(6, 18]$

Step 17