Pre-Algebra Examples

$\frac{1}{3} x + \frac{2}{5} x - \frac{1}{2} \cdot (x + 3) \leq \frac{1}{10}$

Step 1

Simplify $\frac{1}{3} x + \frac{2}{5} x - \frac{1}{2} \cdot (x + 3)$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Combine $\frac{1}{3}$ and $x$ .

$\frac{x}{3} + \frac{2}{5} x - \frac{1}{2} \cdot (x + 3) \leq \frac{1}{10}$

Step 1.1.2

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

$\frac{x}{3} + \frac{2 x}{5} - \frac{1}{2} \cdot (x + 3) \leq \frac{1}{10}$

Step 1.1.3

Apply the distributive property.

$\frac{x}{3} + \frac{2 x}{5} - \frac{1}{2} x - \frac{1}{2} \cdot 3 \leq \frac{1}{10}$

Step 1.1.4

Combine $x$ and $\frac{1}{2}$ .

$\frac{x}{3} + \frac{2 x}{5} - \frac{x}{2} - \frac{1}{2} \cdot 3 \leq \frac{1}{10}$

Step 1.1.5

Multiply $- \frac{1}{2} \cdot 3$ .

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Step 1.1.5.1

Multiply $3$ by $- 1$ .

$\frac{x}{3} + \frac{2 x}{5} - \frac{x}{2} - 3 (\frac{1}{2}) \leq \frac{1}{10}$

Step 1.1.5.2

Combine $- 3$ and $\frac{1}{2}$ .

$\frac{x}{3} + \frac{2 x}{5} - \frac{x}{2} + \frac{- 3}{2} \leq \frac{1}{10}$

Step 1.1.6

Move the negative in front of the fraction.

$\frac{x}{3} + \frac{2 x}{5} - \frac{x}{2} - \frac{3}{2} \leq \frac{1}{10}$

Step 1.2

To write $\frac{x}{3}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{x}{3} \cdot \frac{5}{5} + \frac{2 x}{5} - \frac{x}{2} - \frac{3}{2} \leq \frac{1}{10}$

Step 1.3

To write $\frac{2 x}{5}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{x}{3} \cdot \frac{5}{5} + \frac{2 x}{5} \cdot \frac{3}{3} - \frac{x}{2} - \frac{3}{2} \leq \frac{1}{10}$

Step 1.4

Write each expression with a common denominator of $15$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.4.1

Multiply $\frac{x}{3}$ by $\frac{5}{5}$ .

$\frac{x \cdot 5}{3 \cdot 5} + \frac{2 x}{5} \cdot \frac{3}{3} - \frac{x}{2} - \frac{3}{2} \leq \frac{1}{10}$

Step 1.4.2

Multiply $3$ by $5$ .

$\frac{x \cdot 5}{15} + \frac{2 x}{5} \cdot \frac{3}{3} - \frac{x}{2} - \frac{3}{2} \leq \frac{1}{10}$

Step 1.4.3

Multiply $\frac{2 x}{5}$ by $\frac{3}{3}$ .

$\frac{x \cdot 5}{15} + \frac{2 x \cdot 3}{5 \cdot 3} - \frac{x}{2} - \frac{3}{2} \leq \frac{1}{10}$

Step 1.4.4

Multiply $5$ by $3$ .

$\frac{x \cdot 5}{15} + \frac{2 x \cdot 3}{15} - \frac{x}{2} - \frac{3}{2} \leq \frac{1}{10}$

Step 1.5

Combine the numerators over the common denominator.

$\frac{x \cdot 5 + 2 x \cdot 3}{15} - \frac{x}{2} - \frac{3}{2} \leq \frac{1}{10}$

Step 1.6

Simplify each term.

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Step 1.6.1

Simplify the numerator.

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Step 1.6.1.1

Factor $x$ out of $x \cdot 5 + 2 x \cdot 3$ .

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Step 1.6.1.1.1

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

$\frac{x (5) + 2 x \cdot 3}{15} - \frac{x}{2} - \frac{3}{2} \leq \frac{1}{10}$

Step 1.6.1.1.2

Factor $x$ out of $2 x \cdot 3$ .

$\frac{x (5) + x (2 \cdot 3)}{15} - \frac{x}{2} - \frac{3}{2} \leq \frac{1}{10}$

Step 1.6.1.1.3

Factor $x$ out of $x (5) + x (2 \cdot 3)$ .

$\frac{x (5 + 2 \cdot 3)}{15} - \frac{x}{2} - \frac{3}{2} \leq \frac{1}{10}$

Step 1.6.1.2

Multiply $2$ by $3$ .

$\frac{x (5 + 6)}{15} - \frac{x}{2} - \frac{3}{2} \leq \frac{1}{10}$

Step 1.6.1.3

Add $5$ and $6$ .

$\frac{x \cdot 11}{15} - \frac{x}{2} - \frac{3}{2} \leq \frac{1}{10}$

Step 1.6.2

Move $11$ to the left of $x$ .

$\frac{11 x}{15} - \frac{x}{2} - \frac{3}{2} \leq \frac{1}{10}$

Step 1.7

To write $\frac{11 x}{15}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{11 x}{15} \cdot \frac{2}{2} - \frac{x}{2} - \frac{3}{2} \leq \frac{1}{10}$

Step 1.8

To write $- \frac{x}{2}$ as a fraction with a common denominator, multiply by $\frac{15}{15}$ .

$\frac{11 x}{15} \cdot \frac{2}{2} - \frac{x}{2} \cdot \frac{15}{15} - \frac{3}{2} \leq \frac{1}{10}$

Step 1.9

Write each expression with a common denominator of $30$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.9.1

Multiply $\frac{11 x}{15}$ by $\frac{2}{2}$ .

$\frac{11 x \cdot 2}{15 \cdot 2} - \frac{x}{2} \cdot \frac{15}{15} - \frac{3}{2} \leq \frac{1}{10}$

Step 1.9.2

Multiply $15$ by $2$ .

$\frac{11 x \cdot 2}{30} - \frac{x}{2} \cdot \frac{15}{15} - \frac{3}{2} \leq \frac{1}{10}$

Step 1.9.3

Multiply $\frac{x}{2}$ by $\frac{15}{15}$ .

$\frac{11 x \cdot 2}{30} - \frac{x \cdot 15}{2 \cdot 15} - \frac{3}{2} \leq \frac{1}{10}$

Step 1.9.4

Multiply $2$ by $15$ .

$\frac{11 x \cdot 2}{30} - \frac{x \cdot 15}{30} - \frac{3}{2} \leq \frac{1}{10}$

Step 1.10

Combine the numerators over the common denominator.

$\frac{11 x \cdot 2 - x \cdot 15}{30} - \frac{3}{2} \leq \frac{1}{10}$

Step 1.11

Simplify each term.

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Step 1.11.1

Simplify the numerator.

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Step 1.11.1.1

Factor $x$ out of $11 x \cdot 2 - x \cdot 15$ .

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Step 1.11.1.1.1

Factor $x$ out of $11 x \cdot 2$ .

$\frac{x (11 \cdot 2) - x \cdot 15}{30} - \frac{3}{2} \leq \frac{1}{10}$

Step 1.11.1.1.2

Factor $x$ out of $- x \cdot 15$ .

$\frac{x (11 \cdot 2) + x (- 1 \cdot 15)}{30} - \frac{3}{2} \leq \frac{1}{10}$

Step 1.11.1.1.3

Factor $x$ out of $x (11 \cdot 2) + x (- 1 \cdot 15)$ .

$\frac{x (11 \cdot 2 - 1 \cdot 15)}{30} - \frac{3}{2} \leq \frac{1}{10}$

Step 1.11.1.2

Multiply $11$ by $2$ .

$\frac{x (22 - 1 \cdot 15)}{30} - \frac{3}{2} \leq \frac{1}{10}$

Step 1.11.1.3

Multiply $- 1$ by $15$ .

$\frac{x (22 - 15)}{30} - \frac{3}{2} \leq \frac{1}{10}$

Step 1.11.1.4

Subtract $15$ from $22$ .

$\frac{x \cdot 7}{30} - \frac{3}{2} \leq \frac{1}{10}$

Step 1.11.2

Move $7$ to the left of $x$ .

$\frac{7 x}{30} - \frac{3}{2} \leq \frac{1}{10}$

Step 2

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

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Step 2.1

Add $\frac{3}{2}$ to both sides of the inequality.

$\frac{7 x}{30} \leq \frac{1}{10} + \frac{3}{2}$

Step 2.2

To write $\frac{3}{2}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{7 x}{30} \leq \frac{1}{10} + \frac{3}{2} \cdot \frac{5}{5}$

Step 2.3

Write each expression with a common denominator of $10$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.3.1

Multiply $\frac{3}{2}$ by $\frac{5}{5}$ .

$\frac{7 x}{30} \leq \frac{1}{10} + \frac{3 \cdot 5}{2 \cdot 5}$

Step 2.3.2

Multiply $2$ by $5$ .

$\frac{7 x}{30} \leq \frac{1}{10} + \frac{3 \cdot 5}{10}$

Step 2.4

Combine the numerators over the common denominator.

$\frac{7 x}{30} \leq \frac{1 + 3 \cdot 5}{10}$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Multiply $3$ by $5$ .

$\frac{7 x}{30} \leq \frac{1 + 15}{10}$

Step 2.5.2

Add $1$ and $15$ .

$\frac{7 x}{30} \leq \frac{16}{10}$

Step 2.6

Cancel the common factor of $16$ and $10$ .

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Step 2.6.1

Factor $2$ out of $16$ .

$\frac{7 x}{30} \leq \frac{2 (8)}{10}$

Step 2.6.2

Cancel the common factors.

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Step 2.6.2.1

Factor $2$ out of $10$ .

$\frac{7 x}{30} \leq \frac{2 \cdot 8}{2 \cdot 5}$

Step 2.6.2.2

Cancel the common factor.

$\frac{7 x}{30} \leq \frac{2 \cdot 8}{2 \cdot 5}$

Step 2.6.2.3

Rewrite the expression.

$\frac{7 x}{30} \leq \frac{8}{5}$

Step 3

Multiply both sides by $30$ .

$\frac{7 x}{30} \cdot 30 \leq \frac{8}{5} \cdot 30$

Step 4

Simplify.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Cancel the common factor of $30$ .

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Step 4.1.1.1

Cancel the common factor.

$\frac{7 x}{30} \cdot 30 \leq \frac{8}{5} \cdot 30$

Step 4.1.1.2

Rewrite the expression.

$7 x \leq \frac{8}{5} \cdot 30$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $\frac{8}{5} \cdot 30$ .

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Step 4.2.1.1

Cancel the common factor of $5$ .

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Step 4.2.1.1.1

Factor $5$ out of $30$ .

$7 x \leq \frac{8}{5} \cdot (5 (6))$

Step 4.2.1.1.2

Cancel the common factor.

$7 x \leq \frac{8}{5} \cdot (5 \cdot 6)$

Step 4.2.1.1.3

Rewrite the expression.

$7 x \leq 8 \cdot 6$

Step 4.2.1.2

Multiply $8$ by $6$ .

$7 x \leq 48$

Step 5

Divide each term in $7 x \leq 48$ by $7$ and simplify.

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Step 5.1

Divide each term in $7 x \leq 48$ by $7$ .

$\frac{7 x}{7} \leq \frac{48}{7}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $7$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{7 x}{7} \leq \frac{48}{7}$

Step 5.2.1.2

Divide $x$ by $1$ .

$x \leq \frac{48}{7}$

Step 6

The result can be shown in multiple forms.

Inequality Form:

$x \leq \frac{48}{7}$

Interval Notation:

$(- \infty, \frac{48}{7}]$